Лекции по алгебре. 1 семестр

‹¥ªæ¨¨ ¯®  «£¥¡à¥, I ᥬ¥áâà, ¬¥å-¬ â Œƒ“

‚. €. €àâ ¬®­®¢

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‘Ž„…†€ˆ…

ƒ‹€‚€ 1

‘¨áâ¥¬ë «¨­¥©­ëå ãà ¢­¥­¨© ¨ ¬ âà¨æë

1. Œ¥â®¤ ƒ ãáá 

 áᬮâਬ ¯àאַ㣮«ì­ãî á¨á⥬㠫¨­¥©­ëå ãà ¢­¥­¨©

ª®íää¨æ¨¥­âë ª®â®à®©

  a11 x1 + · · · + a1n xn = b1 .....................................  am1 x1 + · · · + amn xn = bm aij , bj

(1)

§ ¤ ­ë.  è  楫ì { ­ ©â¨ ¢á¥ à¥è¥­¨ï ¨ 㪠§ âì  «£®à¨â¬

¤«ï ­ å®¦¤¥­¨ï ¢á¥å à¥è¥­¨©. Ž¯à¥¤¥«¥­¨¥

çâ® ¤«ï ¢á¥å

1.1. ¥è¥­¨¥¬ á¨á⥬ë (1) ­ §ë¢ ¥âáï â ª®© ­ ¡®à ç¨á¥«

i = 1, . . . , m

(α1 , . . . , αn ),

¢ë¯®«­¥­ë à ¢¥­á⢠

ai1 α1 + · · · + ain αn = bi . Œë ¡ã¤¥¬ à §«¨ç âì á«¥¤ãî騥 ¢¨¤ë á¨á⥬ (1). Ž¯à¥¤¥«¥­¨¥

1.2. ‘¨á⥬  (1) ­¥á®¢¬¥áâ­  , ¥á«¨ ®­  ­¥ ¨¬¥¥â à¥è¥­¨ï. ‘¨á⥬  (1)

ᮢ¬¥áâ­  , ¥á«¨ ®­  ¨¬¥¥â à¥è¥­¨¥. ‘®¢¬¥áâ­ ï á¨á⥬  (1) ­¥®¯à¥¤¥«¥­­  , ¥á«¨ ®­  ¨¬¥¥â ¡®«¥¥ ®¤­®£® à¥è¥­¨ï. ‘®¢¬¥áâ­ ï á¨á⥬  (1) ®¯à¥¤¥«¥­­  , ¥á«¨ ®­  ¨¬¥¥â ¥¤¨­á⢥­­®¥ à¥è¥­¨¥. Œë ¡ã¤¥¬ ᮢ¥àè âì àï¤ ¯à¥®¡à §®¢ ­¨© á¨á⥬ë (1), ­¥ ¬¥­ïî騥 ¬­®¦¥á⢠ ¥¥ à¥è¥­¨©. Ž¯à¥¤¥«¥­¨¥

1.3. „¢¥ á¨áâ¥¬ë «¨­¥©­ëå ãà ¢­¥­¨© ¢¨¤  (1) íª¢¨¢ «¥­â­ë , ¥á«¨

®­¨ ¨¬¥îâ ®¤¨­ ª®¢ë¥ ¬­®¦¥á⢠ à¥è¥­¨©. Œë ¡ã¤¥¬ ᮢ¥àè âì àï¤ ¯à®á⥩è¨å ¯à¥®¡à §®¢ ­¨© á¨á⥬ë (1), á®åà ­ïîé¨å ¬­®¦¥á⢠ à¥è¥­¨©. ‡ ¬¥â¨¬, çâ® ¢á¥ ¨­ä®à¬ æ¨ï ® á¨á⥬¥ (1) ᮤ¥à¦¨âáï ¢ ¥¥ â ¡«¨æ¥ ¥¥ ª®íää¨æ¨¥­â®¢. Ž¯à¥¤¥«¥­¨¥

1.4. Œ âà¨æ¥© á¨á⥬ë (1) ­ §ë¢ ¥âáï ¯àאַ㣮«ì­ ï â ¡«¨æ 



 a11 · · · a1n . . . . . . . . . . . . . . . am1 · · · amn

(2)

 áè¨à¥­­®© ¬ âà¨æ¥© á¨á⥬ë (1) ­ §ë¢ ¥âáï ¯àאַ㣮«ì­ ï â ¡«¨æ 



 a11 · · · a1n b1 . . . . . . . . . . . . . . . . . . . . am1 · · · amn bm ‡ ¬¥ç ­¨¥

1.5. ˆ­®£¤  à áè¨à¥­­ãî ¬ âà¨æã á¨á⥬ë (1) ®¡®§­ ç îâ ç¥à¥§



a11  ··· am1

··· ··· ···

a1n ··· amn 5

 b1 ··· . bm

(3)

6

1. ‘ˆ‘’…Œ› ‹ˆ…‰›• “€‚…ˆ‰ ˆ Œ€’ˆ–›

Ž¯à¥¤¥«¥­¨¥

1.6. ‘«¥¤ãî騥 ¯à¥®¡à §®¢ ­¨ï á¨á⥬ë (1) (¥¥ (à áè¨à¥­­®©) ¬ â-

à¨æë) ­ §ë¢ îâáï í«¥¬¥­â à­ë¬¨ :

♦

¯à¨¡ ¢«¥­¨¥ ª ®¤­®¬ã ãà ¢­¥­¨î (áâப¥) ¤à㣮£® ãà ¢­¥­¨ï (¤à㣮© áâப¨), 㬭®¦¥­­®£®(®©) ­  ¯à®¨§¢®«ì­®¥ ç¨á«®;

♥

㬭®¦¥­¨¥ ãà ¢­¥­¨¥ (áâப¨) ­  ­¥­ã«¥¢®¥ ç¨á«®.

’¥®à¥¬ 

1.7. à¨ í«¥¬¥­â à­ëå ¯à¥®¡à §®¢ ­¨ïå ¯¥à¥å®¤¨¬ ª íª¢¨¢ «¥­â­®© á¨á-

⥬¥. „®ª § â¥«ìá⢮.

­®© á¨á⥬ë (1). ¯®¤áâ ¢¨¬ ­ ¡®à

♦, ¨¬¥­­®,

à¥¤¯®«®¦¨¬, çâ® ¬ë ᮢ¥àè ¥¬ ¯à¥®¡à §®¢ ­¨¥ ⨯ 

ª i-®¬ã ãà ¢­¥­¨î ¯à¨¡ ¢«ï¥¬

j -®¥,

㬭®¦¥­­®¥ ­ 

α.

ɇǬ

‚ᥠãà ¢­¥­¨ï ­®¢®© á¨á⥬ë, ªà®¬¥

(β1 , . . . , βn )

(β1 , . . . , βn )

i-£®,

{ à¥è¥­¨¥ ¨á室-

­¥ ¨§¬¥­¨«¨áì.

ɇǬ ‘

¢ i-®¥ ãà ¢­¥­¨¥ ­®¢®© á¨á⥬ë, â® ¯®«ã稬

(ai1 + αaj1 )β1 + · · · + (ain + αajn )βn = (ai1 β1 + · · · + ain βn ) + α(aj1 β1 + · · · + ajn βn ) = bi + α + bj . ’ ª¨¬ ®¡à §®¬,

(β1 , . . . , βn )

ï¥âáï à¥è¥­¨¥¬ ­®¢®© á¨á⥬ë. ®áª®«ìªã ¨á室­ ï á¨á-

⥬ë (1) ¯®«ãç ¥âáï ¨§ ­®¢®© á¨á⥬ë í«¥¬¥­â à­ë¬ ¯à¥®¡à §®¢ ­¨¥¬ ¯à¨¡ «¥­¨¥¬ ª

i-®¬ã ãà ¢­¥­¨î j -£®, 㬭®¦¥­­®£® ­  −α, â®  ­ «®£¨ç­®, ª ¦¤®¥ à¥è¥­¨¥ ­®¢®© á¨á⥬ë ï¥âáï à¥è¥­¨¥¬ ¨á室­®© á¨á⥬ë. “¯à ¦­¥­¨¥

1.8. „®ª § âì, ç⮠ᮢ¥àè ï í«¥¬¥­â à­ë¥ ¯à¥®¡à §®¢ ­¨ï á® áâப ¬¨

¬ âà¨æë ¬®¦­® ¢ ­¥© ¯¥à¥áâ ¢¨âì «î¡ë¥ ¤¢¥ áâப¨. ã¤¥¬ ¯à¨¢®¤¨âì ¬ âà¨æã á¨áâ¥¬ë ª ­ ¨¡®«¥¥ ¯à®á⮬ã { áâ㯥­ç â®¬ã ¢¨¤ã. Ž¯à¥¤¥«¥­¨¥

1.9. Œ âà¨æ  (3) ­ §ë¢ ¥âáï áâ㯥­ç â®© , ¥á«¨

(1) ­¨¦¥ ­ã«¥¢®© áâப¨ à á¯®«®¦¥­ë ⮫쪮 ­ã«¥¢ë¥ áâப¨; (2) ¯¥à¢ë© ­¥­ã«¥¢®© ª ¦¤®© áâப¨ à ¢¥­ 1; (3) ¥á«¨ ¯¥à¢ë© ­¥­ã«¥¢®© (a)

i-®©

áâப¨ à á¯®«®¦¥­ ­  ¬¥áâ¥

(i, ki ),

â®

ki+1 > ki ;

(b) ¢á¥ í«¥¬¥­âë ’¥®à¥¬ 

aj,ki = 0

¤«ï ¢á¥å

j 6= i.

1.10. Š ¦¤ ï ¬ âà¨æ  ª®­¥ç­ë¬ ç¨á«®¬ í«¥¬¥­â à­ëå ¯à¥®¡à §®¢ ­¨©

áâப ¯à¨¢®¤¨âáï ª áâ㯥­ç â®¬ã ¢¨¤ã. „®ª § â¥«ìá⢮.

ãáâì ¬ âà¨æ 

A

¨¬¥¥â ¢¨¤ (3). …᫨

A = 0,

â® ®­  㦥 ¨¬¥¥â

áâ㯥­ç âë© ¢¨¤. ãáâì

A 6= 0.

ã¤¥¬ ¢¥á⨠¤®ª § â¥«ìá⢮ ¨­¤ãªæ¨¥© ¯® ç¨á«ã áâப

m. ¥§ ®£à ­¨ç¥ai1 . …᫨ i = 1, = 1. ‘«¥¤®¢ â¥«ì­®,

­¨ï ®¡é­®á⨠¬®¦­® áç¨â âì, çâ® ¢ ¯¥à¢®¬ á⮫¡æ¥ ¥áâì ­¥­ã«¥¢®© í«¥¬¥­â ⮠㬭®¦¨¬ 1-ãî áâப㠭 

a−1 11 .

ˆâ ª, ¬®¦­® ¯à¥¤¯®« £ âì, çâ® a11 m = 1, ⮠⥮६  ¤®ª § ­ . ãáâì m > 1, ¨ ¤«ï m − 1 ⥮६  ¤®ª § ­ . „«ï ª ¦¤®£® i > 1 ¢ëç⥬ ¨§ i-®© áâப¨ −1 ¯¥à¢ãî áâபã, 㬭®¦¥­­ãî ­  ai1 a11 . ‚ ­®¢®© ¬ âà¨æ¥ ¢á¥ ª®íää¨æ¨¥­âë ai1 = 0, i > 1.  áᬮâਬ ¯®¤¬ âà¨æã B ¢ A, ¯®«ãç îéãîáï ®â¡à á뢠­¨¥¬ ¯¥à¢®© áâப¨. ® ¨­¤ãªæ¨¨ ¬®¦­® áç¨â âì, çâ® ¬ âà¨æ  B ¨¬¥¥â áâ㯥­ç âë© ¢¨¤. ãáâì ¢ ¬ âà¨æ¥ B ¯¥à¢ë¥ ­¥­ã«¥¢ë¥ í«¥¬¥­âë à á¯®«®¦¥­ë ¢ á⮫¡æ å á ­®¬¥à ¬¨ 1 < k2 < k3 < · · · . ‚ëç⥬ ¨§ ¯¥à¢®© áâப¨ 2-ãî áâபã, 㬭®¦¥­­ãî ­  a1,k2 , âà¥âìî áâபã 3-ãî áâபã, 㬭®¦¥­­ãî ­  a1,k3 , ¨ â. ¤. ¥á«¨

Ž¯à¥¤¥«¥­¨¥

­¥¨§¢¥áâ­ãî

xi

1.11. ãáâì ¬ âà¨æ  á¨á⥬ë (1) ¨¬¥¥â áâ㯥­ç âë© ¢¨¤.

£« ¢­®© , ¥á«¨ ¢ ­¥ª®â®à®¬ ãà ¢­¥­¨¨ ¢á¥ ª®íää¨æ¨¥­âë ¯à¨

à ¢­ë ­ã«ï,   ª®íää¨æ¨¥­â ¯à¨ ­¥¨§¢¥áâ­ë¥ ­ §®¢¥¬ ᢮¡®¤­ë¬¨ .

xi

 §®¢¥¬

x1 , . . . , xi−1

®â«¨ç¥­ ®â ­ã«ï (¨ ¯®â®¬ã à ¢¥­ 1). ¢á¥ ®áâ «ì­ë¥

2. Œ€’ˆ–› ˆ Ž…€–ˆˆ €„ ˆŒˆ

7

à¨¬¥­¨¬ ⥮६ë 1.7, 1.10 ª ¨áá«¥¤®¢ ­¨î á¨á⥬ë (1). ‚ ᨫã 㪠§ ­­ëå ⥮६ ¬®¦­® áç¨â âì, çâ® à áè¨à¥­­ ï ¬ âà¨æ  á¨á⥬ë (1) ¨¬¥¥â áâ㯥­ç âë© ¢¨¤. ãáâì ¥¥ ¯®á«¥¤­ïï ­¥­ã«¥¢ ï áâப  ¨¬¥¥â ¢¨¤ (4)

(0, . . . , 0, 1). â® ®§­ ç ¥â, çâ® á¨á⥬ë (1) ᮤ¥à¦¨â ãà ¢­¥­¨¥

0x1 + · · · + 0xn = 1, çâ® ­¥¢®§¬®¦­®. ‘«¥¤®¢ â¥«ì­®, ¢ í⮬ á«ãç ¥ á¨á⥬  ­¥á®¢¬¥áâ­ . ãáâì ¢ £« ¢­ë¥,  

A ­¥â áâப¨ (4). à¥¤¯®«®¦¨¬ ¤«ï ¯à®áâ®âë, çâ® ¯¥à¥¬¥­­ë¥ x1 , . . . , xr xr+1 , . . . , xn ᢮¡®¤­ë¥. ’®£¤  á¨á⥬  ¨¬¥¥â ¢¨¤  x1 +a1,r+1 xr+1 + · · · + a1n xn = b1    x2 +a2,r+1 xr+1 + · · · + a2n xn = b2 (5) . . . . . . . . . . . . . . .............................    xr +ar,r+1 xr+1 + · · · + arn xn = br

¥à¥­®áï ᢮¡®¤­ë¥ ¯¥à¥¬¥­­ë¥ ¢ ¯à ¢ãî ç áâì, ¯®«ãç ¥¬ ¢ëà ¦¥­¨¥ £« ¢­ëå ­¥¨§¢¥áâ­ëå ç¥à¥§ ᢮¡®¤­ë¥

 x1 = b1 − a1,r+1 xr+1 − · · · − a1n xn    x2 = b2 − a2,r+1 xr+1 − · · · − a2n xn ......................................    xr = br − ar,r+1 xr+1 − · · · − arn xn .

(6)

’ ª¨¬ ®¡à §®¬, ¯à¨¤ ¢ ï ᢮¡®¤­ë¬ ­¥¨§¢¥áâ­ë¬ ¯à®¨§¢®«ì­ë¥ §­ ç¥­¨ï, ¬ë ®¤­®§­ ç­® ­ å®¤¨¬ §­ ç¥­¨ï £« ¢­ëå ­¥¨§¢¥áâ­ëå. ˆâ ª, á¨á⥬  ᮢ¬¥áâ­ , ¨, ¥á«¨ ¥áâì ᢮¡®¤­ë¥ ­¥¨§¢¥áâ­ë¥, â® á¨á⥬  ­¥®¯à¥¤¥«¥­­ . …᫨ ¢á¥ ­¥¨§¢¥áâ­ë¥ £« ¢­ë¥, â® á¨á⥬  ®¯à¥¤¥«¥­­ . Ž¯à¥¤¥«¥­¨¥

¥.

1.12. ‘¨á⥬  (1) ®¤­®à®¤­  , ¥á«¨ ¢á¥ ¥¥ ᢮¡®¤­ë¥ ç«¥­ë ­ã«¥¢ë¥, â.

b1 = · · · = bm = 0. 1.13. …᫨ ¢ ®¤­®à®¤­®© á¨á⥬¥ ç¨á«® ­¥¨§¢¥áâ­ëå

à¥¤«®¦¥­¨¥

ãà ¢­¥­¨©

m,

n

¡®«ìè¥ ç¨á« 

â® á¨á⥬  ­¥®¯à¥¤¥«¥­­ .

„®ª § â¥«ìá⢮.

à¨¢¥¤¥¬ á¨á⥬㠪 áâ㯥­ç â®¬ã ¢¨¤ã. Ÿá­®, çâ® á­®¢  ¯®«ã稬

®¤­®à®¤­ãî á¨á⥬ã, ¯à¨ç¥¬ ç¨á«® £« ¢­ëå ­¥¨§¢¥áâ­ëå ­¥ ¯à¥¢®á室¨â ç¨á«  ­¥­ã«¥¢ëå ãà ¢­¥­¨©, â. ¥. ­¥ ¢á¥ ­¥¨§¢¥áâ­ë¥ £« ¢­ë¥. 2. Œ âà¨æë ¨ ®¯¥à æ¨¨ ­ ¤ ­¨¬¨ Ž¯à¥¤¥«¥­¨¥

¨

m

á⮫¡æ ¬¨.

Mat(n × m) { ¢á¥å ¬ âà¨æ (¯àאַ㣮«ì­ëå â ¡«¨æ) á n áâப ¬¨ A ∈ Mat(n × m), â® ¬ë ¡ã¤¥¬ â ª¦¥ ¯¨á âì A = An×m . …᫨ = (bij ), â® ¯®« £ ¥¬ A + B = (aij + bij ). Šà®¬¥ ⮣®, λAn×m = (λaij ).

1.14.

ɇǬ

An×m = (aij ), Bn×m à¥¤«®¦¥­¨¥

1.15. ãáâì

A, B, C ∈ Mat(n × m)

¨

λ, ν

{ ç¨á« . ’®£¤  á¯à ¢¥¤«¨¢ë

á«¥¤ãî騥 8  ªá¨®¬ ¢¥ªâ®à­®£® ¯à®áâà ­á⢠: (1) (2)

A + B = B + A; A + (B + C) = (A + B) + C ;

(3) ¥á«¨ 0 { ­ã«¥¢ ï ¬ âà¨æ  (¢á¥ ¥¥ ª®íää¨æ¨¥­âë à ¢­ë ­ã«î), â®

«î¡®© ¬ âà¨æë

(4) ¤«ï «î¡®© ¬ âà¨æë (5) (6) (7) (8)

A+0=A

A;

λ(A + B) = λA + λB ; (λ + ν)A = λA + νA; (λν)A = λ(νA); 1A = A.

A

áãé¥áâ¢ã¥â â ª ï ¬ âà¨æ 

−A,

çâ®

A + (−A) = 0;

¤«ï

8

1. ‘ˆ‘’…Œ› ‹ˆ…‰›• “€‚…ˆ‰ ˆ Œ€’ˆ–›

à¨¢¥¤¥¬, ­ ¯à¨¬¥à, ¤®ª § â¥«ìá⢮ ¯¥à¢®£® ã⢥ত¥­¨ï, …᫨

„®ª § â¥«ìá⢮.

A = (aij ), B = (bij ),

â®

A + B = (aij + bij ) = (bij + aij ) = B + A.

Žáâ «ì­ë¥ ã⢥ত¥­¨ï

¤®ª §ë¢ îâáï  ­ «®£¨ç­®î Ž¯à¥¤¥«¥­¨¥

1.16. ãáâì

An×m = (aij ), Cm×k = (cst ). ’®£¤ 

D = AC ∈ Mat(n × k) = (dis ),

£¤¥ ¤«ï ¢á¥å

i = 1, . . . , n,

s = 1, . . . , k

dis = ai1 d1s + · · · + ain dns à¥¤«®¦¥­¨¥

(7)

1.17. “¬­®¦¥­¨¥ ¬ âà¨æ  áá®æ¨ â¨¢­®, â.¥.

(AC)F = A(CF )

¤«ï

«î¡ëå ¬ âà¨æ

A ∈ Mat(n × m), C ∈ Mat(m × k), F ∈ Mat(k × l). „®ª § â¥«ìá⢮.

ãáâì

A = An×m = (aij ), …᫨

C = Cm×k = (cst ),

F = Fk×l = (ftq ).

D ¨§ ®¯à¥¤¥«¥­¨ï 1.16, â® ¯® (7) ­  ¬¥á⥠(i, q) ¢ ¬ âà¨æ¥ (AC)F = DF Pk =1 di, f,q = Pk Pn =1 =1 ai, d, f,q .

á⮨â í«¥¬¥­â

(8)

‘ ¤à㣮© áâ®à®­ë, ¥á«¨ â® ­  ¬¥áâ¥

(i, q)

¢ ¬ âà¨æ¥

CF = U = (ui, ) ∈ Mat(m × l), A(CF ) = AU á⮨â í«¥¬¥­â Pn =1 ai, u,q = Pn Pk =1 ai, d, f,q . =1

(9)

ˆ§ (8), (9) ¢ë⥪ ¥â ã⢥ত¥­¨¥. à¥¤«®¦¥­¨¥

(1) (2)

1.18. ‘¯à ¢¥¤«¨¢ë à ¢¥­á⢠:

λ(AB) = (λA)B = A(λB). A(B + C) = AB + AC, (A + U )V = AV + U V.

„®ª § â¥«ìá⢮.

(bij ), C = (cij ).

„®ª ¦¥¬, ­ ¯à¨¬¥à, ¢â®à®¥ ã⢥ত¥­¨¥. ãáâì

’®£¤  ­  ¬¥áâ¥

X

aik (bkj

¢ ¬ âà¨æ¥

ª®â®àë© à ¢¥­ í«¥¬¥­âã, áâ®ï饬㠭  â® ¦¥ ¬¥á⥠¢ ¬ âà¨æ¥ ¬ âà¨æ

A(B +C) ¨ AB +AC

A = (aij ), B =

A(B + C) áâநâ í«¥¬¥­â X X + ckj ) = aik bkj + aik ckj ,

(i, j)

AB + AC .

’ ª ª ª à §¬¥àë

ᮢ¯ ¤ îâ, â® ®­¨ à ¢­ë. €­ «®£¨ç­® ¯à®¢¥àïîâáï ®áâ «ì­ë¥

ã⢥ত¥­¨ï. Ž¯à¥¤¥«¥­¨¥ à¥¤«®¦¥­¨¥

1.19. ãáâì

1.20. ãáâì

„®ª § â¥«ìá⢮.

á⮨â

Pn

j=1

aij bji ,

A = (aij ) ∈ Mat(n).

ãáâì

A, B ∈ Mat(n).

‘«¥¤®¬

’®£¤ 

A = (aij ), B = (bij ).

tr A ­ §ë¢ ¥âáï a11 + · · · + ann .

tr(AB) = tr(BA).

’®£¤  ­  ¬¥áâ¥

®âªã¤ 

tr(AB) =

n X

aij bji .

i,j=1

€­ «®£¨ç­®,

tr(BA) =

n X

s,t=1

bst ats =

n X

s,t=1

ats bst = tr(AB).

(i, i)

¢ ¬ âà¨æ¥

AB

2. Œ€’ˆ–› ˆ Ž…€–ˆˆ €„ ˆŒˆ

1.21. ‘¨¬¢®« Šà®­¥ª¥à 

Ž¯à¥¤¥«¥­¨¥

­¨ç­ ï ¬ âà¨æ  Šà®­¥ª¥à 

E = En ∈ Mat(n)

δij

à ¢¥­ 1, ¥á«¨

i = j,

9

¨ 0, ¥á«¨

{ íâ® ¬ âà¨æ , ¢ ª®â®à®© ­  ¬¥áâ¥

(i, j)

i 6= j .

…¤¨-

á⮨â ᨬ¢®«

δij . 1.22. ãáâì

à¥¤«®¦¥­¨¥

„®ª § â¥«ìá⢮.

ãáâì

A ∈ Mat(n × m).

A = (aij ). n X

’®£¤ 

En A = A = AEm .

’®£¤  ­  ¬¥áâ¥

(i, j)

¢ ¬ âà¨æ¥

En A

á⮨â

δik akj = δii aij = aij ,

k=1

â. ¥.

En A = A. 1.23. ãáâì

Ž¯à¥¤¥«¥­¨¥

∗

A ∈ Mat(m × n)

1.24. t (A + B)

„®ª § â¥«ìá⢮.

(i, j)

á⮨â

t t

C A.

’®£¤  â࠭ᯮ­¨à®¢ ­­ ï ¬ âà¨æ  t A

{ íâ® ¬ âà¨æ , ¢ ª®â®à®© ­  ¬¥áâ¥

à¥¤«®¦¥­¨¥

¬¥áâ¥

A ∈ Mat(n × m).

P

k

= t A + t B,

t

(i, j)

á⮨â í«¥¬¥­â

(λA) = λt A,

t

aji

¬ âà¨æë

=

A.

(AC) = t Ct A.

„®ª ¦¥¬, ­ ¯à¨¬¥à, ¯®á«¥¤­¥¥ ã⢥ত¥­¨¥. ‚ ¬ âà¨æ¥ t (AC) ­ 

ajk cki =

P

k cki ajk , â. ¥. í«¥¬¥­â, áâ®ï饩 ­  ⮬ ¦¥ ¬¥á⥠¢ ¬ âà¨æ¥

€­ «®£¨ç­® ¤®ª §ë¢ îâáï ®áâ «ì­ë¥ ã⢥ত¥­¨ï. 1.25. Œ âà¨ç­ë¥ ¥¤¨­¨æë

Ž¯à¥¤¥«¥­¨¥

ª®â®àëå ­  ¬¥áâ¥

(s, t)

á⮨â í«¥¬¥­â

δsi δtj ,

Eij ∈ Mat(n × m) { íâ® (i, j) á⮨â 1,

â. ¥. ­  ¬¥áâ¥

¬ âà¨æë

Eij ,

¢

¨ ¢á¥ ®áâ «ì­ë¥

í«¥¬¥­âë à ¢­ë 0. “¯à ¦­¥­¨¥

1.26. „®ª § âì, çâ®

t

♠ Eij = Eji ; ♣ ¥á«¨ A = (aij ), à¥¤«®¦¥­¨¥

â®

1.27.

A=

P

i,j

aij Eij .

Eij Ers = δjr Eis .

Eij Ers á⮨â í«¥¬¥­â ( X 1, u = i = p = j, v = s; (δui δpj)(δpi δvs ) = 0 ¢ ¯à®â¨¢­®¬ á«ãç ¥. p

„®ª § â¥«ìá⢮.

  ¬¥áâ¥

(u, v)

¢

Žâá ¢ë⥪ ¥â ã⢥ত¥­¨¥. ‘«¥¤á⢨¥

1.28. ãáâì

A = (ars ) ∈ Mat(n × m).

’®£¤ 

Eij A = aj1 Ei1 + · · · + ajm Eim , AEij = a1i E1j + · · · + anj Eni . ’¥®à¥¬ 

1.29. —â®¡ë ¢ ¬ âà¨æ¥

㬭®¦¥­­ãî ­ 

α

A ∈ Mat(n × m) ª i-®© áâப¥ (En + αEij )A.

¯à¨¡ ¢¨âì

j -ãî,

­ã¦­® à áᬮâà¥âì ¬ âà¨æã

„®ª § â¥«ìá⢮.

® ¯à¥¤«®¦¥­¨ï¬ 1.18, 1.27, ã¯à ¦­¥­¨ï¬ 1.22,

♣ ¨ á«¥¤á⢨î 1.28

(En + αEij )A = En A + αEij A = P A + α(aj1 Ei1 + · · · + ajm Eim ) = rs ars Ers + (αaj1 )Ei1 + · · · + (αajm )Eim = P P r6=i,s ars Ers + i,s (ais + αajs )Eis . ‘«¥¤á⢨¥

j -ë©,

1.30. —â®¡ë ¢ ¬ âà¨æ¥

㬭®¦¥­­ãî ­ 

Ž¡®§­ ç¥­¨¥

α

A ∈ Mat(n × m) ª i-®¬ã á⮫¡æã A(Em + αEji ).

­ã¦­® à áᬮâà¥âì ¬ âà¨æã

1.31. ®«®¦¨¬

Di (α) = En + (α − 1)Eii ∈ Mat(n).

¯à¨¡ ¢¨âì

10

1. ‘ˆ‘’…Œ› ‹ˆ…‰›• “€‚…ˆ‰ ˆ Œ€’ˆ–›

’¥®à¥¬ 

­ 

α

1.32. —â®¡ë ¢ ¬ âà¨æ¥

­ã¦­® à áᬮâà¥âì ¬ âà¨æã

A ∈ Mat(n × m) i-ãî Di (α)A (ADi (α)).

áâபã (á⮫¡¥æ) 㬭®¦¨âì

ãáâì A = (aij ). ‡ ¬¥â¨¬, çâ® ¢ ¬ âà¨æ¥ Di (α) ­  δst + (α − 1)δis δit . ®í⮬㠢 ¬ âà¨æ¥ Di (α)A ­  ¬¥á⥠(p, q) á⮨â P P P r (δpr + (α − 1)δip δir )arq = r δpr arq + (α − 1) r δip δir arq = ( apq , i 6= q; apq + (α − 1)δip aiq = αaiq , p = i.

„®ª § â¥«ìá⢮.

á⮨â

‡ ¬¥ç ­¨¥

¬¥áâ¥

(s, t)

1.33. ‚ â¥à¬¨­ å ¬ âà¨ç­®£® 㬭®¦¥­¨ï 㤮¡­® § ¯¨á뢠âì á¨áâ¥¬ë «¨-

­¥©­ëå ãà ¢­¥­¨©. ˆ¬¥­­®, á¨á⥬ë (1) ¨¬¥¥â ¢¨¤ (1),



 X=

x1 . . .

xn

  

{ á⮫¡¥æ ­¥¨§¢¥áâ­ëå,



 b=

b1 . . .

bm

{ á⮫¡¥æ ᢮¡®¤­ëå ç«¥­®¢.

  

AX = b,

£¤¥

A

{ ¬ âà¨æ  (2) á¨á⥬ë

ƒ‹€‚€ 2

Œ­®¦¥á⢠ ¨ ®â®¡à ¦¥­¨ï. ¥à¥áâ ­®¢ª¨

1. Œ­®¦¥á⢠ ¨ ®â®¡à ¦¥­¨ï.

2.1. Žâ®¡à ¦¥­¨¥

Ž¯à¥¤¥«¥­¨¥

f : X → Y.

(10)

Žâ®¡à ¦¥­¨¥ (10) (1) ¨­ê¥ªâ¨¢­® , ¥á«¨ ¤«ï «î¡ëå

x, y ∈ X ¨§ f (x) = f (y) á«¥¤ã¥â x = y ; y ∈ Y áãé¥áâ¢ã¥â â ª®¥ x ∈ X , çâ® f (x) = y ;

(2) áîàꥪ⨢­® , ¥á«¨ ¤«ï «î¡®£®

(3) ¡¨¥ªâ¨¢­® , ¥á«¨ ®­® ¨­ê¥ªâ¨¢­® ¨ áîàꥪ⨢­®. 2.2. ãáâì § ¤ ­® ®â®¡à ¦¥­¨¥ (10). „«ï «î¡®£®

Ž¡®§­ ç¥­¨¥

f “¯à ¦­¥­¨¥

−1

y∈Y

¯®«®¦¨¬

(y) = {x ∈ X|f (x) = y}.

2.3. „®ª § âì, çâ® ®â®¡à ¦¥­¨¥ (10)

|f −1 (y)| ≤ 1 ¤«ï «î¡®£® y ∈ Y ; áîàꥪ⨢­® ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤  |f −1 (y)| ≥ 1 ¤«ï «î¡®£® y ∈ Y ; ¡¨¥ªâ¨¢­® ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤  |f −1 (y)| = 1 ¤«ï «î¡®£® y ∈ Y .

(1) ¨­ê¥ªâ¨¢­® ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤  (2) (3)

2.4. ãáâì

Ž¯à¥¤¥«¥­¨¥

gf : X → Z . : Y → X.

¦¥­¨©

f −1

à¥¤«®¦¥­¨¥

f : X → Y, g : Y → Z . à®¨§¢¥¤¥­¨¥ (ª®¬¯®§¨æ¨ï) ®â®¡à 1X : X → X . Ž¡à â­®¥ ®â®¡à ¦¥­¨¥

’®¦¤¥á⢥­­®¥ ®â®¡à ¦¥­¨¥

2.5. ‘¯à ¢¥¤«¨¢ë á«¥¤ãî騥 ã⢥ত¥­¨ï:

(1) 㬭®¦¥­¨¥ ®â®¡à ¦¥­¨©  áá®æ¨ â¨¢­®; (2) ¯à®¨§¢¥¤¥­¨¥ ¨­ê¥ªâ¨¢­ëå ®â®¡à ¦¥­¨© ¨­ê¥ªâ¨¢­®; (3) ¯à®¨§¢¥¤¥­¨¥ áîàꥪ⨢­ëå ®â®¡à ¦¥­¨© áîàꥪ⨢­®; (4) ¥á«¨ (5) ¥á«¨

f f

¨§ (10), â®

1Y f = f 1X = f ; f −1 f = 1X ¨ f f −1 = 1Y ; ®â®¡à ¦¥­¨¥ ª f ¨§ (10) áãé¥áâ¢ã¥â

¨§ (10), â®

(6) ®¡à â­®¥

⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤ 

f

¡¨¥ªâ¨¢­®. “¯à ¦­¥­¨¥

2.6. Œ­®¦¥á⢮

X ª®­¥ç­® ⮣¤  ¨ ⮫쪮 X → X ¡¨¥ªâ¨¢­®.

⮣¤ , ª®£¤  «î¡®¥ ¨­ê¥ª-

⨢­®¥ (áîàꥪ⨢­®¥) ®â®¡à ¦¥­¨¥

2. ¥à¥áâ ­®¢ª¨

ãáâì

Xn = {1, 2, . . . , n}.

Ž¯à¥¤¥«¥­¨¥

®â®¡à ¦¥­¨¥

Xn

2.7. ¥à¥áâ ­®¢ª®© (¯®¤áâ ­®¢ª®©) á⥯¥­¨

¢ ᥡï. —¥à¥§

à¥¤«®¦¥­¨¥

Sn

n ­ §ë¢ ¥âáï ¡¨¥ªâ¨¢­®¥ n.

®¡®§­ ç ¥âáï ¬­®¦¥á⢮ ¢á¥å ¯¥à¥áâ ­®¢®ª á⥯¥­¨

2.8. à®¨§¢¥¤¥­¨¥ ¯¥à¥áâ ­®¢®ª ¨ ®¡à â­ ï ¨ ⮦¤¥á⢥­­ ï ¯¥à¥-

áâ ­®¢ª¨ á­®¢  ïîâáï ¯¥à¥áâ ­®¢ª ¬¨. “¬­®¦¥­¨¥ ¯¥à¥áâ ­®¢®ª  áá®æ¨ â¨¢­®. „®ª § â¥«ìá⢮.

ã¦­® ¢®á¯®«ì§®¢ âìáï ¯à¥¤«®¦¥­¨¥¬ 2.5. 11

12

2. ŒŽ†…‘’‚€ ˆ Ž’Ž€†…ˆŸ. ……‘’€Ž‚Šˆ

Ž¡®§­ ç¥­¨¥

2.9. ãáâì

σ ∈ Sn .

’®£¤  ¥á«¨

Xn = {i1 , . . . , in },

â®

σ

®¤­®§­ ç­®

§ ¤ ¥âáï ¢ ¢¨¤¥ ¤¢ãáâà®ç­®© ¬ âà¨æë

σ= “¯à ¦­¥­¨¥

2.10. ãáâì

σ



 in σ(i1 )

i1 ... σ(i1 ) . . .

(11)

¨§ (11), ¨



τ=

j1 i1

... ...

 jn . in

’®£¤ 



στ =

j1 ... σ(i1 ) . . .

 jn σ(in )

¨

σ Ž¯à¥¤¥«¥­¨¥

Sn

¤«¨­ë

k

„¢  横« 

−1

=

2.11. ãáâì i1 , . . .



σ(i1 ) . . . i1 ...

–¨ª«®¬

(i1 , . . . , ik ) ∈

­¥§ ¢¨á¨¬ë , ¥á«¨ ¢á¥ í«¥¬¥­âë i1 , . . .

, ik , j1 , . . . , js

, ik

{ à §«¨ç­ë¥ ç¨á«  ¨§

­ §ë¢ ¥âáï â ª ï ¯¥à¥áâ ­®¢ª 

  is+1 , σ(m) = i1 ,   m, ,

(i1 , . . . , ik ), (j1 , . . . , js ) ∈ Sn

 σ(in ) i1

σ,

¥á«¨ ¥á«¨ ¥á«¨

çâ® ¤«ï

Xn .

m ∈ Xn

m = is , s < k; m = ik ; m ∈ Xn \ {i1 , . . . , ik }.

à §«¨ç­ë. ’¥®à¥¬ 

2.12. ‹î¡ ï ¯¥à¥áâ ­®¢ª  à §« £ ¥âáï ¢ ¯à®¨§¢¥¤¥­¨¥ ­¥§ ¢¨á¨¬ëå 横-

«®¢. ãáâì σ ∈ Sn . Œ®¦­® áç¨â âì, çâ® σ 6= 1. ‚®§ì¬¥¬ ¯à®¨§¢®«ì­ë© k, 1 ≤ k ≤ n, ¨ ¯à¥¤¯®«®¦¨¬, çâ® í«¥¬¥­âë k0 = k, k1 = σk, k2 = σ 2 k, . . . , kl = σ l k à §«¨ç­ë, ­® σ l+1 k = σ s k , £¤¥ 0 ≤ s ≤ l. „®ª § â¥«ìá⢮.

í«¥¬¥­â

‹¥¬¬ 

2.13.

s = 0.

…᫨ s > 0, â® σ(ks−1 ) = σ(kl ), çâ® ­¥¢®§¬®¦­®, X = {1, . . . , n}, ­® ks−1 6= kl ¢ á¨«ã ¢ë¡®à  l.

„®ª § â¥«ìá⢮.

¨­ê¥ªâ¨¢­® ­ 

ˆâ ª, ­  ¬­®¦¥á⢥

{k0 , k1 , . . . , kl } ¯®¤áâ ­®¢ª  σ ¤¥©áâ¢ã¥â   k0 k1 . . . kl−1 kl k1 k2 . . . kl k0

‹¥¬¬ 

¤¥©áâ¢ã¥â

j, 1 ≤ j ≤ n, ¯à¨ç¥¬ j ∈ / {k0 , k1 , . . . , kl }. Š ª {j0 , j1 , . . . , jt }, ­  ª®â®à®¬ ¯®¤áâ ­®¢ª  σ ¤¥©áâ¢ã¥â ª ª 横«   j0 j1 . . . jt−1 jt j1 j2 . . . jt j0

2.14. ‚á¥ í«¥¬¥­âë

„®ª § â¥«ìá⢮.

ãáâì

k0 , k1 , . . . , kl , j0 , j1 , . . . , jt

jr = kq . j0 = σ

çâ® ­¥¢®§¬®¦­®.

σ

ª ª

‚롥६ ⥯¥àì ¯à®¨§¢®«ì­®¥ ç¨á«® áâந¬ ¬­®¦¥á⢮

¨¡®

−r

à §«¨ç­ë.

’®£¤ 

jr ∈ {k0 , k1 , . . . , kl },

¨ ¢ëè¥

2. ……‘’€Ž‚Šˆ

13

à®¤®«¦ ï íâ®â ¯à®æ¥áá, ¯®«ãç ¥¬ ¯®¤áâ ­®¢ªã

τ=



k0 k1

k1 k2

... ...

kl−1 kl

kl k0

 j0 j1

¥¯®á।á⢥­­ ï ¯à®¢¥àª  ¯®ª §ë¢ ¥â, çâ® à¥¤«®¦¥­¨¥

2.15. ãáâì

¨

π ∈ Sn

π(i1 , . . . , ik )π „®ª § â¥«ìá⢮. Ž¯à¥¤¥«¥­¨¥ ’¥®à¥¬ 

j1 j2

 jt ··· . j0

jt−1 jt

τ = σ.

(i1 , . . . , ik )

−1

... ...

{ 横« ¨§

Sn .

’®£¤ 

= (π(i1 ), . . . , π(ik )).

¥¯®á।á⢥­­ ï ¯à®¢¥àª .

2.16. ’࠭ᯮ§¨æ¨¥© ­ §ë¢ ¥âáï 横« ¤«¨­ë 2.

2.17. Š ¦¤ ï ¯¥à¥áâ ­®¢ª  ï¥âáï ¯à®¨§¢¥¤¥­¨¥¬ â࠭ᯮ§¨æ¨©.

„®ª § â¥«ìá⢮. Ž¯à¥¤¥«¥­¨¥

(i1 , . . . , ik ) = (i1 , i2 )(i2 , i3 ) · · · (ik−1 , ik ).

2.18. ãáâì

i 1 , . . . , in

{ ¯®á«¥¤®¢ â¥«ì­®áâì à §«¨ç­ëå ç¨á¥« ¨§

ˆ­¢¥àᨥ© ¢ í⮩ ¯®á«¥¤®¢ â¥«ì­®á⨠­ §ë¢ ¥âáï â ª ï ¯ à  ª®¬ ¯®á«¥¤®¢ â¥«ì­®á⨠­ §ë¢ ¥âáï ç¨á«®

is , it , çâ® s < t ¨ is > it .

Xn . ‡­ -

(−1)M ,

£¤¥ M { ç¨á«® ¨­¢¥àᨩ ¢ ¯®á«¥¤®¢ σ ∈ Sn ¨¬¥¥â ¤¢ãáâà®ç­ãî § ¯¨áì (11), £¤¥ i1 = 1, . . . , in = n, σ à ¢¥­ §­ ªã ¯®á«¥¤®¢ â¥«ì­®á⨠¨§ ¢â®à®© áâப¨.

⥫쭮áâ¨. …᫨ ¯®¤áâ ­®¢ª  â® §­ ª

(−1)σ

’¥®à¥¬ 

¯¥à¥áâ ­®¢ª¨

2.19. ãáâì § ¤ ­  ¯¥à¥áâ ­®¢ª  (11), £¤¥

ik = k

¤«ï ¢á¥å

k.

à¥¤¯®«®-

¦¨¬, çâ® § ¤ ­ë à §«¨ç­ë¥ ç¨á« 

y1 , . . . , yn . ’®£¤  Y yσt − yσs . (−1)sigma = yt − y s

(12)

1≤s 0. ‚롥६ â ª®¥ ­ ¨¬¥­ì襥 ­ âãà «ì­®¥ ç¨á«® m, çâ® b = am ∈ H . …᫨ ar ∈ H, r ∈ Z, â®, ¤¥«ï r á ®áâ âª®¬ ­  m, ¯®«ãç ¥¬ r = sm + q, 0 ≤ q < m. à¨ í⮬ ¯® ¯à¥¤«®¦¥­¨î 6.14 aq = ar−sm = ar (am )−s ∈ H, çâ® ¯à®â¨¢®à¥ç¨â ¢ë¡®à㠑«¥¤á⢨¥

„®ª § â¥«ìá⢮.

H = Zd.

’¥®à¥¬ 

„®ª § â¥«ìá⢮.

d = (n, k).

G = hain ¨ H { ¯®¤£à㯯  ¢ G. d, ¤¥«ï饥 n, çâ® H = had i nd .

6.26. ãáâì

H = ha i,

à¨¬¥à

¨, á«¥¤®-

’®£¤ 

«¥¬¥­â

ak

|ak | =

0 ≤ k < n.

n . (n, k)

ï¥âáï ¯®à®¦¤ î騬 ¢

¤«ï ¢á¥å

x, y ∈ G.

‚ ª ç¥á⢥

f

â® ®¯à¥¤¥«¨¬

G = hain .

f :G→H '.

¢§ïâì

­ §ë¢ ¥âáï ¨§®¬®à-

exp.

n ¨§®¬®àä­  Un .

f : G → Un , 2πi f (ak ) = exp( ). n f : G → Z, ¯®« £ ï f (ak ) = k .

Zn ' Un .

‡ ¤ ¤¨¬

⮣¤ 

Ž¡®§­ ç¥­¨¥

Z.

ãáâì

G

hai12 .

6.31. –¨ª«¨ç¥áª ï £à㯯  ¯®à浪 

6.32.

¤«ï ­¥ª®â®à®£®

G = hai.  áᬮâਬ ¯®¤£à㯯ã H = hak i ⊆ d = (n, k). Žâá |ak | = |H| = nd .

(R, +) ' (R>0 , ·).

„®ª § â¥«ìá⢮.

‘«¥¤á⢨¥

n.

6.29. ¨¥ªâ¨¢­®¥ ®â®¡à ¦¥­¨¥ £à㯯

à¥¤«®¦¥­¨¥

G = hai∞ ,

¨¬¥¥â ¯®à冷ª

G = hain . (k, n) = 1.

«¨ç¥áª ï £à㯯  ¨§®¬®àä­ 

ɇǬ

£¤¥

f (xy) = f (x)f (y)

6.30.

H = hak i H = had i.

6.28. Ž¯¨á âì ¢á¥ ¯®¤£àã¯¯ë ¢

Ž¯à¥¤¥«¥­¨¥

䨧¬®¬ , ¥á«¨

a∈G

6.27. ãáâì

¨ ⮫쪮 ⮣¤ , ª®£¤  “¯à ¦­¥­¨¥

Z,

Œ®¦­® áç¨â âì, çâ® d

® ⥮६¥ 6.25 ‘«¥¤á⢨¥

{ ¯®¤£à㯯  ¢

’®£¤  áãé¥áâ¢ã¥â ¨ ¯à¨â®¬

® ⥮६¥ 6.23 ¯®«ãç ¥¬

Žáâ ¥âáï § ¬¥â¨âì, çâ®

„®ª § â¥«ìá⢮.

G.

H = Zm1 + · · · + Zmn . ’®£¤  H d = (m1 , . . . , mn ).

ãáâì

’®£¤  áãé¥áâ-

Žáâ ¥âáï ã¡¥¤¨âìáï, çâ®

6.25. ãáâì

‘«¥¤á⢨¥

q > 0.

m1 , . . . , mn ∈ Z , ¨ d { ç¨á¥« m1 , . . . , mn . u1 , . . . , un ∈ Z , çâ® m1 u1 + · · · + mn un = d.

¥¤¨­á⢥­­®¥ â ª®¥ ç¨á«®

®«®¦¨¬

¥á«¨

6.24. ãáâì

¢ãîâ â ª¨¥ æ¥«ë¥ ç¨á« 

¢ â¥«ì­®,

m,

¯®« £ ï

¥áª®­¥ç­ ï 横-

36

6. ƒ“›, ŠŽ‹œ–€ ˆ Ž‹Ÿ

2. ‘¬¥¦­ë¥ ª« ááë ¨ ⥮६  ‹ £à ­¦ 

6.33. ãáâì

Ž¯à¥¤¥«¥­¨¥

ª« áᮬ

gH

{ ¯®¤£à㯯  ¢ £à㯯¥

H

­ §ë¢ ¥âáï ¯®¤¬­®¦¥á⢮

“¯à ¦­¥­¨¥

{gh|h ∈ H}

G,

¨

‹¥¢ë¬ ᬥ¦­ë¬

g ∈ G.

G.

6.34.  ©â¨

(1) «¥¢ë¥ ᬥ¦­ë¥ ª« ááë

GL(n, C) Z ¯® nZ;

¯®

(3) «¥¢ë¥ ¨ ¯à ¢ë¥ ᬥ¦­ë¥ ª« ááë

Sn

(2) «¥¢ë¥ ᬥ¦­ë¥ ª« ááë

“¯à ¦­¥­¨¥

¢

6.35. ãáâì

SL(n, C); ¯®

Sn−1 .

{ ¯®¤£à㯯  ¢ £à㯯¥

H

G

¨

x, y ∈ G.

G

¨

„®ª § âì, çâ® á«¥¤ã-

î騥 ãá«®¢¨ï íª¢¨¢ «¥­â­ë: (1) (2)

xH = yH ; x−1 y ∈ H .

à¥¤«®¦¥­¨¥

6.36. ãáâì

H

{ ¯®¤£à㯯  ¢ £à㯯¥

à¥¤«®¦¥­¨¥

6.37. ãáâì

H

{ ¯®¤£à㯯  ¢ £à㯯¥

’®£¤ 

„®ª § â¥«ìá⢮.

â. ¥.

x, y ∈ G,

|H| = |xH|.

¯à¨ç¥¬

y ∈ xH .

Ÿá­®, çâ®

yH ⊆ xH .

u∈H

¯®«ãç ¥¬

® ãá«®¢¨î y = xh ¤«ï ­¥ª®â®à®£® h ∈ H . xu = y(h−1 u), £¤¥ h−1 u ∈ H . Žâá xH ⊆

xH = yH .

‘«¥¤á⢨¥

­ëå ª« áá 

G

¯®

6.38. ãáâì

H

’¥®à¥¬ 

|G| = |H|j ,

£¤¥

{ ¯®¤£à㯯  ¢ £à㯯¥

j

G.

’®£¤  ¤¢  «¥¢ëå (¯à ¢ëå) ᬥ¦-

‚®á¯®«ì§®¢ âìáï ¯à¥¤«®¦¥­¨¥¬ 6.37.

6.39 (’¥®à¥¬  ‹ £à ­¦ ) . ãáâì

H

{ ¯®¤£à㯯  ¢ ª®­¥ç­®© £à㯯¥

{ ç¨á«® «¥¢ëå (¯à ¢ëå) ᬥ¦­ëå ª« áá 

„®ª § â¥«ìá⢮.

x∈G

H

«¨¡® ᮢ¯ ¤ îâ, «¨¡® ­¥ ¯¥à¥á¥ª îâáï.

„®ª § â¥«ìá⢮.

¬¥­â

¨

’®£¤ 

xH = yH .

‘«¥¤®¢ â¥«ì­®, ¤«ï «î¡®£®

yH ,

G

x ∈ G.

 §®¡ê¥¬

G

G

¯®

­  «¥¢ë¥ ᬥ¦­ë¥ ª« ááë ¯®

«¥¦¨â ¢ ­¥ª®â®à®¬ ª« áá¥, ¨¬¥­­®, ¢

xH .

G.

’®£¤ 

H. H.

’®£¤  ª ¦¤ë© í«¥-

Žáâ ¥âáï ¢®á¯®«ì§®¢ âìáï á«¥¤áâ-

¢¨¥¬ 6.38 ¨ ¯à¥¤«®¦¥­¨¥¬ 6.36. ‘«¥¤á⢨¥

6.40. ®à冷ª í«¥¬¥­â  ª®­¥ç­®© £àã¯¯ë ¤¥«¨â ¯®à冷ª £à㯯ë.

‘«¥¤á⢨¥

6.41. ƒà㯯  ¯à®á⮣® ¯®à浪  ï¥âáï 横«¨ç¥áª®©.

ƒ‹€‚€ 7

Š®«ìæ  ¨ ¯®«ï

Ž¯à¥¤¥«¥­¨¥

7.1. Š®«ìæ® (­¥ ®¡ï§ â¥«ì­®  áá®æ¨ â¨¢­®¥). €áá®æ¨ â¨¢­ë¥, ª®¬¬ã-

â â¨¢­ë¥ ª®«ìæ . à¥¤«®¦¥­¨¥ à¨¬¥àë

♥ ♥

7.2. ‚ «î¡®¬ ª®«ìæ® ¨¬¥¥¬

0x = x0 = 0.

7.3. “ª ¦¥¬ àï¤ ª®«¥æ.

€áá®æ¨ â¨¢­ë¥ ª®«ìæ  { ª®«ìæ  ¬ âà¨æ

Mat(n, R).

€áá®æ¨ â¨¢­®-ª®¬¬ãâ â¨¢­ë¥ ª®«ìæ  { ª®«ìæ  ­¥¯à¥à뢭ëå ä㭪権 ­  ⮯®«®£¨ç¥áª®¬ ¯à®áâà ­á⢥.

Ž¯à¥¤¥«¥­¨¥

7.4. …¤¨­¨ç­ë© í«¥¬¥­â, ¤¥«¨â¥«¨ ­ã«ï , ®¡à â¨¬ë¥ í«¥¬¥­âë  «£¥¡àë.

à¥¤«®¦¥­¨¥

7.5. …¤¨­¨ç­ë© í«¥¬¥­â  «£¥¡àë ®¯à¥¤¥«¥­ ®¤­®§­ ç­®. Ž¡à â¨¬ë¥

í«¥¬¥­âë  áá®æ¨ â¨¢­®©  «£¥¡àë ®¡à §ãîâ £à㯯㠯® 㬭®¦¥­¨î.

Ž¡à â¨¬ë© í«¥-

¬¥­â  áá®æ¨ â¨¢­®©  «£¥¡àë ­¥ ¬®¦¥â ¡ëâì ¤¥«¨â¥«¥¬ ­ã«ï. ‘«¥¤á⢨¥ ’¥®à¥¬ 

­ã«ï ¢

7.6. ‚ ¯®«¥ ­¥â ¤¥«¨â¥«¥© ­ã«ï.

7.7. ƒàã¯¯ë ®¡à â¨¬ëå í«¥¬¥­â®¢ ¢

Mat(n, k)

Ž¯à¥¤¥«¥­¨¥ à¥¤«®¦¥­¨¥ ’¥®à¥¬ 

«¥¬¥­â

7.8. ‘âàãªâãà  ª®«ìæ  ­ 

{ íâ®

GL(n, k);

¤¥«¨â¥«¨

Zn .

7.9. ‘âàãªâãà  ª®«ìæ  ­ 

7.10. «¥¬¥­â

k ∈ Zn

Mat(n, k)

{ íâ® ¢ë஦¤¥­­ë¥ ¬ âà¨æë ¨ ⮫쪮 ®­¨.

k ∈ Zn

Zn

®¯à¥¤¥«¥­  ª®à४⭮.

®¡à â¨¬ ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤ 

ï¥âáï ¤¥«¨â¥«¥¬ ­ã«ï ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤ 

(k, n) = 1. (k, n) > 1.

«¥¬¥­â k ®¡à â¨¬ ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤  ­ ©¤¥âáï â ª®¥ ku = 1, â. ¥. ¢ Z ku + nv = 1. â® íª¢¨¢ «¥­â­® ⮬ã, çâ® (k, n) = 1. ãáâì í«¥¬¥­â k ∈ Zn ï¥âáï ¤¥«¨â¥«¥¬ ­ã«ï. ’®£¤  ®­ ­¥ ¬®¦¥â ¨¬¥âì ®¡à â­®£® ¯® ¯à¥¤«®¦¥­¨î 7.5. ‘«¥¤®¢ â¥«ì­®, (k, n) > 1. Ž¡à â­®, ¯ãáâì d = (k, n) > 1. ’®£¤  n > m = nd−1 , â. ¥. m 6= 0 ¢ Zn . à¨ í⮬ km = k1 dm = k1 n = 0 ¢ Zn . „®ª § â¥«ìá⢮.

ç¨á«®

u ∈ Zn ,

çâ®

Ž¯à¥¤¥«¥­¨¥ à¨¬¥àë

7.11. ®«¥.

7.12. ®«ï {

Ž¯à¥¤¥«¥­¨¥

Q, R, C.

Šà®¬¥ ⮣®,

7.13. • à ªâ¥à¨á⨪  ¯®«ï

√ Q[i], Q[ 5].

char.

’¥®à¥¬ 

7.14. • à ªâ¥à¨á⨪  ¯®«ï «¨¡® à ¢­  ­ã«î, «¨¡® ¯à®á⮥ ç¨á«®.

’¥®à¥¬ 

7.15. Š®«ìæ® ¢ëç¥â®¢

Zn

ï¥âáï ¯®«¥¬ ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤ 

{ ¯à®á⮥ ç¨á«®. „®ª § â¥«ìá⢮. ‘«¥¤á⢨¥

7.16.

ã¦­® ¢®á¯®«ì§®¢ âìáï ⥮६®© 7.10.

char Zp = p. 37

n

38

7. ŠŽ‹œ–€ ˆ Ž‹Ÿ

f : R → R0 ­ §ë¢ ¥âáï f (x + y) = f (x) + f (y), f (xy) = f (x)f (y). Ž¡®§­ ç¥­¨¥ R ' R0 . 7.17. ¨¥ªâ¨¢­®¥ ®â®¡à ¦¥­¨¥ ª®«¥æ

Ž¯à¥¤¥«¥­¨¥

䨧¬®¬ , ¥á«¨ à¨¬¥à

7.18. „®ª § âì, çâ®

“¯à ¦­¥­¨¥

(1)

f (1) = 1

¨§®¬®à-

Mat(n, Mat(m, R)) ' Mat(nm, R).

7.19. „®ª § âì, çâ® ¯à¨ ¨§®¬®à䨧¬¥

f : R → R0

¨ ®¡à â¨¬ë¥ í«¥¬¥­âë ¯¥à¥å®¤ïâ ¢ ®¡à â¨¬ë¥;

(2) ¤¥«¨â¥«¨ ­ã«ï ¯¥à¥å®¤ïâ ¢ ¤¥«¨â¥«¨ ­ã«ï. 7.20. ®¤¬­®¦¥á⢮

Ž¯à¥¤¥«¥­¨¥

(¯®¤¯®«¥¬) , ¥á«¨ ¯®«¥©

x−1 ∈ K,

à¨¬¥àë

K

¥á«¨

7.21.

Z

ï¥âáï ¯®¤ª®«ì殬 ¢

â®

b = c.

Q, Q

¥á«¨ ¤«ï «î¡ëå

7.23. Ž¡« áâﬨ ïîâáï

“¯à ¦­¥­¨¥

ab = ac,

R

­ §ë¢ ¥âáï ¯®¤ª®«ì殬

á«¥¤ã¥â, çâ®

x + y, xy ∈ K

(¢ á«ãç ¥

ï¥âáï ¯®¤¯®«¥¬ ¢

R, C. Mat(n, Z)

7.22. Š®«ìæ® ¡¥§ ¤¥«¨â¥«¥© ­ã«ï ­ §ë¢ ¥âáï ®¡« áâìî . „à㣨¬¨ á«®-

R ï¥âáï ®¡« áâìî, x = 0, «¨¡® y = 0.

à¨¬¥à

x, y ∈ K

Mat(n, R).

¢ ¬¨, ª®«ìæ® çâ® «¨¡®

¢ ª®«ìæ¥ á 1 (¯®«¥)

x 6= 0.

ï¥âáï ¯®¤ª®«ì殬 ¢ Ž¯à¥¤¥«¥­¨¥

K

ᮤ¥à¦¨â 1 ¨ ¨§ ⮣®, çâ®

7.24. ãáâì

a, b, c

Z,

x, y ∈ R

¨§ ⮣®, çâ®

xy = 0,

á«¥¤ã¥â,

«î¡®¥ ¯®«¥.

{ í«¥¬¥­âë ¨§ ­¥ª®â®à®© ®¡« áâ¨, ¨

a 6= 0.

ɇǬ

ƒ‹€‚€ 8

Œ­®£®ç«¥­ë ¨ àï¤ë ®â ®¤­®© ¯¥à¥¬¥­­®©

1. Š®«ìæ® ¬­®£®ç«¥­®¢ ®â ®¤­®© ¯¥à¥¬¥­­®©

ãáâì

R {  áá®æ¨ â¨¢­®¥ ª®«ìæ® á 1.

 áᬮâਬ ¬­®¦¥á⢮

R[X] ¢á¥å ¯®ç⨠­ã«¥¢ëå

¯®á«¥¤®¢ â¥«ì­®á⥩ (34)

a = (a0 , a1 , . . . ) í«¥¬¥­â®¢ ¨§

a

R.

Ž¯à¥¤¥«¨¬ ¢

R[X] ®¯¥à æ¨î á«®¦¥­¨ï ¯®-ª®®à¤¨­ â­®.

Šà®¬¥ ⮣®, ¥á«¨

¨§ (34),

b = (b0 , b1 , . . . ), â®

c = ab = (c0 , c1 , . . . ), £¤¥ ¤«ï «î¡®£®

k≥0 ck =

k X

(35)

ai bk−i .

i=0

à¥¤«®¦¥­¨¥

⨢­®, â® ¨

R[X]

8.2.

Ž¯à¥¤¥«¥­¨¥ à¥¤«®¦¥­¨¥

­®¥

8.1.

R[X]

ï¥âáï  áá®æ¨ â¨¢­ë¬ ª®«ì殬 á 1. …᫨

R

ª®¬¬ãâ -

ª®¬¬ãâ â¨¢­®.

R[X]

­ §ë¢ ¥âáï ª®«ì殬 ¬­®£®ç«¥­®¢ .

8.3. ‚á¥ í«¥¬¥­âë

(a0 , 0, . . . ) ∈ R[X]

®¡à §ãîâ ¯®¤ª®«ìæ®, ¨§®¬®àä-

R. „®ª § â¥«ìá⢮.

®«®¦¨¬

a 7→ (a, 0, . . . ) ∈ R[X].

8.4. ‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¬ë ¡ã¤¥¬ ®â®¦¤¥á⢫ïâì

Ž¡®§­ ç¥­¨¥

R[X].

ˆ§®¬®à䨧¬ § ¤ ¥âáï ¯® ¯à ¢¨«ã

a ∈ R á (a, 0, . . . ) ∈

X = (0, 1, 0, . . . ).

à¥¤«®¦¥­¨¥

8.5. „«ï «î¡®£®

n≥1

n

X = (0, . . . , 0, 1, 0, . . . ) | {z } n

ɇǬ

a

¨§ (34), â®

a = a0 + a1 X + · · · + an X n , 8.6. ãáâì

Ž¯à¥¤¥«¥­¨¥

deg a = n. a0

a ∈ R[X]

ai ∈ R.

¨§ (36), ¯à¨ç¥¬

­ §ë¢ ¥âáï ᢮¡®¤­ë¬ ç«¥­®¬

a. an

an 6= 0.

(36)

‘⥯¥­ìî

a

­ §ë¢ ¥âáï

­ §ë¢ ¥âáï áâ à訬 ç«¥­®¬

a.

Œ­®£®-

ç«¥­ë ­ã«¥¢®© á⥯¥­¨ ¨ ­ã«¥¢®© ¬­®£®ç«¥­ ­ §ë¢ îâáï ª®­áâ ­â ¬¨ . Œ­®£®ç«¥­ë á® áâ à訬 ª®íää¨æ¨¥­â®¬ 1 ­ §ë¢ îâáï ã­¨â à­ë¬¨ . à¥¤«®¦¥­¨¥

8.7. ‘â à訩 (᢮¡®¤­ë©) ç«¥­ ¯à®¨§¢¥¤¥­¨ï ¬­®£®ç«¥­®¢ à ¢¥­

¯à®¨§¢¥¤¥­¨î áâ àè¨å (᢮¡®¤­ëå) ç«¥­®¢ ᮬ­®¦¨â¥«¥©. ‚ ç áâ­®áâ¨, ®¡à â¨¬ë¬¨ í«¥¬¥­â ¬¨

R[X]

ïîâáï ®¡à â¨¬ë¥ ¢

R

8.8. …᫨ R { ®¡« áâì, â® ¨ R[X] { ®¡« áâì. Šà®¬¥ ⮣®, ¥á«¨ f, g ∈ deg(f g) = deg f + deg g . …᫨ f, g, f + g 6= 0, â® deg(f + g) ≤ max(deg f, deg g).

‘«¥¤á⢨¥

r[X] \ 0,

â®

ª®­áâ ­âë.

39

40

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

2. „¥«¥­¨¥ ¬­®£®ç«¥­®¢

8.9. ‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ®

Ž¡®§­ ç¥­¨¥

í⮬ á«ãç ¥, £à㯯  ®¡à â¨¬ëå í«¥¬¥­â®¢

R∗

R

{ ¯®«¥.

‚

ᮢ¯ ¤ ¥â á ¬ã«ì⨯«¨ª â¨¢­®© £à㯯®©

R[X]

¢á¥å ­¥­ã«¥¢ëå ª®­áâ ­â. ’¥®à¥¬ 

8.10 („¥«¥­¨¥ á ®áâ âª®¬) . ãáâì

f, g ∈ R[X], q, r ∈ R[X], çâ®

é¥áâ¢ãîâ ¨ ¯à¨â®¬ ¥¤¨­á⢥­­ë¥ â ª¨¥ (1) (2)

¯à¨ç¥¬

g 6= 0.

’®£¤  áã-

f = qg + r; r = 0 ¨«¨ deg r < deg g .

„®ª § â¥«ìá⢮. ‘ãé¥á⢮¢ ­¨¥.

ɇǬ

f = 0

¨«¨

deg f < deg g ,

â® ¯®« £ ¥¬

q = 0, r = g . ãáâì deg f = n, ¨ ¤«ï an , bm { áâ à訥 ª®íää¨æ¨¥­âë n. ® ¨­¤ãªæ¨¨ h = q 0 g + r0 , ®âªã¤ 

¬¥­ìè¨å á⥯¥­¥© ⥮६  ¤®ª § ­ . à¥¤¯®«®¦¨¬, çâ®

f, g .

an n−m X g ¬¥­ìè¥ bm an n−m an n−m f =h+ X g = (q 0 + X )g + r0 . bm bm

h=f−

’®£¤  á⥯¥­ì

…¤¨­á⢥­­®áâì.

ãáâì

f = qg + r = q 0 g + r0 , ’®£¤ 

0

0

g(q − q ) = r − r.

ɇǬ

deg r, deg r0 < deg g, 0

q − q 6= 0,

â®

0

r − r 6= 0.

0

¥á«¨ ®­¨ ­¥­ã«¥¢ë¥.

Žâá ¯® á«¥¤á⢨î 8.8

deg g > deg(r − r ) = deg(g(q − q )) = deg g + deg(q − q 0 ) ≥ deg g. 8.11. Œ­®£®ç«¥­

Ž¯à¥¤¥«¥­¨¥

¤¥«¥­¨ï ­ 

g.

0

Œ­®£®ç«¥­

g

¤¥«¨â

q

f,

­ §ë¢ ¥âáï ç áâ­ë¬ ,   ¬­®£®ç«¥­

¥á«¨

r = 0.

Ž¡®§­ ç¥­¨¥

8.12.  ¨¡®«ì訬 ®¡é¨¬ ¤¥«¨â¥«¥¬ ¬­®£®ç«¥­®¢

Ž¯à¥¤¥«¥­¨¥

¨§ ª®â®àëå à ¢­ë ­ã«î, ­ §ë¢ ¥âáï â ª®© ¬­®£®ç«¥­

d,

r

®áâ âª®¬

f

g|f . f1 , . . . , fm ,

­¥ ¢á¥

çâ®

(1)

d|fi , i = 1, . . . , m; d0 ∈ R[X], ¨ d0 |fi , i = 1, . . . , m, â® d0 |d.  ¨¡®«ì訩 ®¡é¨© ¤¥«¨â¥«ì ¬­®£®ç«¥­®¢ f1 , . . . , fm , ®¡®§­ ç ¥âáï «¨¡® (f1 , . . . , fm ), «¨¡® (f1 , . . . , fm ). (2) ¥á«¨

à¥¤«®¦¥­¨¥

8.13.

(f1 , . . . , fm ),

®¯à¥¤¥«¥­ ®¤­®§­ ç­®, á â®ç­®áâìî ¤® ¬­®¦¨-

â¥«ï ­ã«¥¢®© á⥯¥­¨ (­¥­ã«¥¢®© ª®­áâ ­âë). Šà®¬¥ ⮣®,

(f1 , (f2 , . . . , fm )). ˆ§«®¦¨¬  «£®à¨â¬ ¢ª«¨¤  ­ å®¦¤¥­¨ï ­ ¨¡®«ì襣® ®¡é¥£® ¤¥«¨â¥«ï ¤¢ãå ¬­®£®ç«¥­®¢

f, g , g 6= 0.

ã¤¥¬ ¤¥«¨âì á ®áâ âª®¬.

f = q 1 g + r1 , deg r1 < deg g; g = q 2 r1 + r2 , deg r2 < deg r1 ; r 1 = q 3 r 2 + r3 , deg r3 < deg r2 ; ............................................ rk = qk+2 rk+1 + rk+2 , deg rk+2 < deg rk+1 ; rk+1 = qk+3 rk+2 . Žâ¬¥â¨¬, çâ® ¢ (37) ç¨á«®

k

áãé¥áâ¢ã¥â, ¯®áª®«ìªã á⥯¥­¨ ®áâ âª®¢ ã¡ë¢ îâ.

(37)

2. „…‹…ˆ… ŒŽƒŽ—‹…Ž‚

’¥®à¥¬ 

41

(f, g) = rk+2 .

8.14.

„®ª § â¥«ìá⢮.

ˆ§ ¯à¥¤¯®á«¥¤­¥£® à ¢¥­á⢠ ¢ (37) ¢ë⥪ ¥â, çâ®

¤. ®¤­¨¬ ïáì ¢¢¥àå ¯®«ãç ¥¬, çâ®

rk+2 |rk+1 ,

¨ â.

rk+2 |g, rk+2 |f .

d0 |f, d0 |g , â® ¨§ ¯¥à¢®£® à ¢¥­á⢠ ¢ (37) ¯®«ãç ¥¬, çâ® d0 |r1 ,   ¨§ ¢â®à®£® ¢­¨§, ¯®«ãç ¥¬, çâ® d0 |rk+2 .

Ž¡à â­®, ¥á«¨ {

0

d |g .

„¢¨£ ïáì

‘«¥¤á⢨¥

u 1 , . . . , um ,

8.15. ãáâì

d = (f1 , . . . , fm ).

’®£¤  áãé¥áâ¢ãîâ â ª¨¥ ¬­®£®ç«¥­ë

çâ®

d = u1 f1 + · · · + um fm . „®ª § â¥«ìá⢮.

g.

‚ ᨫ㠯।«®¦¥­¨ï 8.13 ¬®¦­® áç¨â âì, çâ®

m = 2, ¨ f1 = f, f2 =

Žáâ ¥âáï ¢®á¯®«ì§®¢ âìáï  «£®à¨â¬®¬ (37). 8.16. Œ­®£®ç«¥­ë

Ž¯à¥¤¥«¥­¨¥ à¥¤«®¦¥­¨¥

f1 , . . . , fm

8.17. Œ­®£®ç«¥­ë

¢§ ¨¬­® ¯à®áâë , ¥á«¨

f1 , . . . , fm ¢§ ¨¬­® ¯à®áâë, u1 , . . . , um , çâ®

ª®£¤  áãé¥áâ¢ãîâ â ª¨¥ ¬­®£®ç«¥­ë

(f1 , . . . , fm ) = 1.

⮣¤  ¨ ⮫쪮 ⮣¤ ,

1 = u1 f1 + · · · + um fm . 8.18. Œ­®£®ç«¥­

Ž¯à¥¤¥«¥­¨¥

p

á⥯¥­¨

p

deg p ≥ 1

­ §ë¢ ¥âáï ­¥¯à¨¢®¤¨¬ë¬ , ¥á«¨

­¥ à §« £ ¥âáï ¢ ¯à®¨§¢¥¤¥­¨¥ ¬­®£®ç«¥­®¢ ¬¥­ì襩 á⥯¥­¨. “¯à ¦­¥­¨¥

8.19. „®ª § âì, çâ®

(1) «î¡®© ¬­®£®ç«¥­ ¯¥à¢®© á⥯¥­¨, (2) ¬­®£®ç«¥­

X 2 + 1 ∈ R[X]

­¥¯à¨¢®¤¨¬. à¥¤«®¦¥­¨¥

«¨¡®

8.20. ãáâì

„®ª § â¥«ìá⢮.

0,

­¥¯à¨¢®¤¨¬ ¨

p ∈ R[X]

f ∈ R[X].

’®£¤  «¨¡®

(p, f ) = 1,

p|f .

«¨¡®

ãáâì

d = (p, f ),

¨

f = f 0 d.

‚ ᨫ㠮¯à¥¤¥«¥­¨ï 8.18 «¨¡®

deg d =

deg d = deg p.

‚ ¯¥à¢®¬ á«ãç ¥ ¯® ¯à¥¤«®¦¥­¨î 8.7 í«¥¬¥­â ãáâì −1 0

f = pc

f

d = 1. deg d = deg p. ’®£¤  p = dc,

d ®¡à â¨¬, â.

¥. ¢ ᨫ㠯।«®¦¥­¨ï 8.13

¬®¦­® áç¨â âì, çâ®

p|f ,

«¨¡®

{ ­¥­ã«¥¢ ï ª®­áâ ­â , â. ¥.

d = pc−1 .

Žâá

8.21. ãáâì

p ∈ R[X]

p|(f g),

£¤¥

® ¯à¥¤«®¦¥­¨î 8.20 ¬®¦­® áç¨â âì, çâ®

¯® ¯à¥¤«®¦¥­¨î 8.17. Žâá

’¥®à¥¬ 

­¥¯à¨¢®¤¨¬ ¨

f, g ∈ R[X].

’®£¤ 

p|g .

„®ª § â¥«ìá⢮.

f u + pv

c

.

à¥¤«®¦¥­¨¥

«¨¡®

£¤¥

g = 1g = f gu + pvg.

®í⮬ã

(p, f ) = 1. p|g .

’®£¤ 

1=

8.22. ‹î¡®© ­¥­ã«¥¢®© ¬­®£®ç«¥­ à §« £ ¥âáï ¢ ¯à®¨§¢¥¤¥­¨¥ ­¥¯à¨¢®¤¨-

¬ëå ¬­®£®ç«¥­®¢. …᫨

f = p 1 · · · pn = q 1 · · · q m

(38)

{ ¤¢  à §«®¦¥­¨ï ¢ ¯à®¨§¢¥¤¥­¨ï ­¥¯à¨¢®¤¨¬ëå ¬­®£®ç«¥­®¢

n = m,

¨ áãé¥áâ¢ã¥â â ª ï ¯¥à¥áâ ­®¢ª 

σ ∈ Sn ,

çâ®

p1 , . . . , pn , q1 , . . . , qm , â® pi = ci qσi , £¤¥ ci { ­¥­ã«¥¢ ï

ª®­áâ ­â . „®ª § â¥«ìá⢮. ‘ãé¥á⢮¢ ­¨¥.

ˆ­¤ãªæ¨ï ¯®

deg f .

‘«ãç ©

deg f = 1

®ç¥¢¨¤¥­, â ª ª ª ¢ í⮬ á«ãç ¥

ãáâì ¤«ï ¬¥­ìè¨å á⥯¥­¥© ã⢥ত¥­¨¥ ¤®ª § ­®. …᫨

f

f

­¥¯à¨¢®¤¨¬.

­¥¯à¨¢®¤¨¬, â® ã⢥ত¥­¨¥

42

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

¤®ª § ­®. …᫨

f = gh,

£¤¥

deg g, deg g < deg f ,

â® ­ã¦­® ¢®á¯®«ì§®¢ âìáï ¨­¤ãªæ¨¥© ¤«ï

g, h. ˆ­¤ãªæ¨ï ¯®

…¤¨­á⢥­­®áâì.

deg f .

‘«ãç ©

deg f = 1

®ç¥¢¨¤¥­. ãáâì ¤«ï ¬¥­ì-

è¨å á⥯¥­¥© ã⢥ত¥­¨¥ ¤®ª § ­®, ¨ ¢ë¯®«­¥­® (38). ® ¯à¥¤«®¦¥­¨î 8.21 ­¥ª®â®à®£® ¯®«ãç ¥¬

i. ¥à¥­ã¬¥à®¢ë¢ ï, ¬®¦­® áç¨â âì, çâ® i = 1. q1 = c1 p1 , £¤¥ c1 { ­¥­ã«¥¢ ï ª®­áâ ­â . ’®£¤ 

p1 |qi

¤«ï

‚ ᨫ㠭¥¯à¨¢®¤¨¬®áâ¨

q1

p1 (p2 · · · pn − (c1 q2 )q3 · · · qm ) = 0. ® ¯à¥¤«®¦¥­¨î 8.8 ¨ ã¯à ¦­¥­¨î 7.24 ¯®«ãç ¥¬, çâ®

p2 · · · pn − (c1 q2 )q3 · · · qm = 0, ¨«¨

p2 · · · pn = (c1 q2 )q3 · · · qm . ¯® ¨­¤ãªæ¨¨

n−1=m−1

Ž¯à¥¤¥«¥­¨¥

¨

pi = ci qi , i ≥ 2, ci ∈ R∗ .

8.23. ‚ ᨫã ⥮६ë 8.22 ª ¦¤ë© ­¥­ã«¥¢®© ¬­®£®ç«¥­

f ∈ R[X]

®¤-

­®§­ ç­® ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥

c ∈ R∗ ,

f = pl11 · · · plnn , £¤¥

pi

¢

p1 , . . . , pn f. p

(39)

{ ã­¨â à­ë¥ ­¥¯à¨¢®¤¨¬ë¥ ¬­®£®ç«¥­ë. —¨á«®

8.24. ãáâì

’¥®à¥¬ 

¬­®£®ç«¥­ 

li ∈ N ∪ 0,

¢

f,

char R = 0, ¨ f ∈ R[X] \ 0. p ¢ f 0 à ¢­  k − 1.

ɇǬ

k

li

­ §ë¢ ¥âáï ªà â­®áâìî

{ ªà â­®áâì ­¥¯à¨¢®¤¨¬®£®

â® ªà â­®áâì ˆ¬¥¥¬

„®ª § â¥«ìá⢮.

f = pk g ,

£¤¥

(p, g) = 1.

’®£¤ 

f 0 = kpk−1 p0 g + pk g = pk−1 (kp0 g + pg 0 ).

(40)

pk−1 |f 0 . …᫨ pk |f 0 , â® ¯® p|(kp0 g + pg 0 ), ¨ ¯®í⮬ã p|kp0 g . p|kp0 , «¨¡® p|g , ® ¢â®à®¥ ­¥¢®§¬®¦­® ¯® ¯à¥¤¯®«®¦¥­¨î,   ¯à¥¤¯®«®¦¥­¨î kp0 6= 0 ¨ deg(kp0 ) < deg p.

‘«¥¤®¢ â¥«ì­®,

® ¯à¥¤«®¦¥-

­¨î 8.21 «¨¡®

¯¥à¢®¥ ­¥¢®§-

¬®¦­®, ¨¡® ¯®

‘«¥¤á⢨¥

8.25. ãáâì

char R = 0,

¨

f ∈ R[X] \ 0

d = (f, f 0 ) = pl11 −1 · · · plnn −1 ,

¨

¨¬¥¥â à §«®¦¥­¨¥ (39). ’®£¤ 

f = p1 · · · p n . d

3. Š®à­¨ ¬­®£®ç«¥­®¢ Ž¯à¥¤¥«¥­¨¥

· · · + an cn .

8.26. ãáâì

R

{ ¯®«¥ ¨

«¥¬¥­â ï¥âáï ª®à­¥¬

“¯à ¦­¥­¨¥

8.27. ãáâì

a,

c ∈ R. „«ï a a(c) = 0.

f, g ∈ R[X].

’¥®à¥¬ 

8.28 (’¥®à¥¬  ¥§ã) . «¥¬¥­â

Žâá

(f g)(c) = f (c)g(c). c ∈ R

ï¥âáï ª®à­¥¬

f ∈ R[X]

⮣¤  ¨

(X − c)|f .

„®ª § â¥«ìá⢮.

r ∈ R.

a(c) = a0 + a1 c +

’®£¤ 

(f + g)(c) = f (c) + g(c), ⮫쪮 ⮣¤ , ª®£¤ 

¨§ (36) ¯®«®¦¨¬

¥á«¨

r = f (c).

„¥«ï

f

á ®áâ âª®¬ ­ 

X −c

¯®«ãç ¥¬, çâ®

f = (X − c)q + r,

£¤¥

3. ŠŽˆ ŒŽƒŽ—‹…Ž‚

Žâ¬¥â¨¬, çâ® ¥á«¨

X − c,

¯®«ãç ¥¬ ¤«ï

f

c ∈ R,

43

â® ¤¥«ï «î¡®© ¬­®£®ç«¥­ ¯®á«¥¤®¢ â¥«ì­® á ®áâ âª®¬ ­ 

à §«®¦¥­¨¥ ’¥©«®à 

f = b0 + b1 (X − c) + · · · + bn (X − c)n .

(41)

ˆ§«®¦¨¬ á奬㠃®à­¥à  ¤«ï ¡ëáâண® ¢ëç¨á«¥­¨ï ª®íää¨æ¨¥­â®¢

¢ (41) ¯® ª®íää¨æ¨-

¥­â ¬

ãáâì

f.

„«ï í⮣® ­ã¦­® 㬥âì ¤¥« âì ®¤¨­ è £ ¤¥«¥­¨ï ­  n

f = a0 + a1 X + · · · + an X = g(X − c) + r,

bi X − c.

r ∈ R,

(42)

£¤¥

g = s0 + s1 X + · · · + sn−1 X n−1 . ®¤áâ ¢«ïï ¢ (42) ¯®«ãç ¥¬

a0 + a1 X + · · · + an X n = (X − c)(s0 + s1 X + · · · + sn−1 X n−1 ) + r. à¨à ¢­¨¢ ï ª®íää¨æ¨¥­âë ¯à¨ ®¤¨­ ª®¢ëå á⥯¥­ïå

X,

¯®«ãç ¥¬

an = sn−1 an−1 = sn−2 − csn−1 ........................ a1 = s0 − cs1 a0 = r − cs0 . Žâá

sn−1 = an sn−2 = an−1 + csn−1 ........................ s0 = a1 + cs1 r = a0 + cs0 . ”®à¬ã«ë (43) ¯®§¢®«ïîâ ¡ëáâà® § 

(43)

㬭®¦¥­¨© ¢ëç¨á«¨âì

n

r = f (c).

¥§ã«ìâ âë íâ¨å

¢ëç¨á«¥­¨© ®¡ëç­® § ¯¨á뢠îâáï ¢ ¢¨¤¥ â ¡«¨æë

an c a0 = sn−1 „¥«ï ¤ «¥¥

g

an−1 sn−2

X − c ¯®«ãç ¥¬,

á ®áâ âª®¬ ­ 

... ...

a1 s1

a0 r

(44)

¯®«ãç ¥¬ ä®à¬ã«ã ’¥©«®à . ¥§ã«ìâ âë íâ¨å

¢ëç¨á«¥­¨© § ¯¨è¥¬ ¢ â ¡«¨æã

an c an = sn−1 c an = tn−2

£¤¥

bi

an−1 sn−2 tn−3

... ... ... . . .

. . .

. . .

. . .

c c

an = u1 an = bn

u0 = bn−1

a1 s1 t 0 = b1

a0 r = b0 (45)

,

¨§ (41).

Ž¯à¥¤¥«¥­¨¥

8.29. Šà â­®áâìî ª®à­ï

¢®¤¨¬®£® ¬­®¦¨â¥«ï k+1

(X − c)

X −c

¢

f.

c ¬­®£®ç«¥­  f

­ §ë¢ ¥âáï ªà â­®áâì ­¥¯à¨-

„à㣨¬¨ á«®¢ ¬¨, ªà â­®áâì à ¢­ 

k,

¥á«¨

- f.

â® ®¯à¥¤¥«¥­¨¥ ᮣ« á®¢ ­® á ⥮६®© ¥§ã 8.28 “¯à ¦­¥­¨¥

8.30. ‘«¥¤ãî騥 ãá«®¢¨ï íª¢¨¢ «¥­â­ë:

(1) ªà â­®áâì ª®à­ï

c

¬­®£®ç«¥­ 

f

à ¢­ 

k;

(2) ¢ ä®à¬ã«¥ ’¥©«®à 

f = bk (X − c)k + bk+1 (X − c)k+1 + · · · + bn (X − c)n ,

bk 6= 0.

(X − c)k |f ,

­®

44

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

ɇǬ

c1 , . . . , cm

{ à §«¨ç­ë¥ ª®à­¨

f

á ªà â­®áâﬨ k1

f = (X − c1 )

k1 , . . . , km , km

· · · (X − cm )

â®

g.

®í⮬ã á¯à ¢¥¤«¨¢® 8.31. ‘㬬  ç¨á«  ª®à­¥© (á ªà â­®áâﬨ) ­¥ ¯à¥¢®á室¨â á⥯¥­¨

à¥¤«®¦¥­¨¥

¬­®£®ç«¥­ . 8.32. ãáâì

“¯à ¦­¥­¨¥

R { ¯®«¥ å à ªâ¥à¨á⨪¨ 0. «¥¬¥­â c ∈ R ï¥âáï ªà â≥ 2) ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤  c ®¡é¨© ª®à¥­ì ¬­®£®-

­ë¬ ª®à­¥¬ (ª®à­¥¬ ªà â­®á⨠童­  ¨ ¥£® ¯à®¨§¢®¤­®©.

4. ˆ­â¥à¯®«ïæ¨ï

ãáâì

a0 , a − 1, . . . , an ,

{ ¯®«¥ á à §«¨ç­ë¬¨ í«¥¬¥­â ¬¨

R

¨

b0 , b − 1, . . . , bn ∈ R.

 áᬮâਬ ¨­â¥à¯®«ï樮­­ë© ¬­®£®ç«¥­ ‹ £à ­¦ 

F =

n X

bi

i=0

’¥®à¥¬ 

8.33.

(X − a0 ) · · · (X − ai−1 )(X − ai+1 ) · · · (X − an ) . (ai − a0 ) · · · (ai − ai−1 )(ai − ai+1 ) · · · (ai − an ) ¨

deg F ≤ n

F (aj ) = bj

¤«ï ¢á¥å

(46)

j = 0, . . . , n.

„®ª § â¥«ìá⢮.

F (aj ) =

n X

bi

i=0

’¥®à¥¬ 

F (aj ) = bj

(aj − a0 ) · · · (aj − ai−1 )(aj − ai+1 ) · · · (aj − an ) = bj . (ai − a0 ) · · · (ai − ai−1 )(ai − ai+1 ) · · · (ai − an )

8.34. ‘ãé¥áâ¢ã¥â ¢ â®ç­®á⨠®¤¨­ ¬­®£®ç«¥­

¤«ï ¢á¥å

„®ª § â¥«ìá⢮.

G = F1 − F2 6= 0,

F

á⥯¥­¨

≤n

á ãá«®¢¨¥¬

j = 0, . . . , n. ãáâì áãé¥áâ¢ãîâ ¤¢  ¬­®£®ç«¥­ 

deg G ≤ n,

¯à¨ç¥¬

¨ í«¥¬¥­âë

F1 , F2 á í⨬ ãá«®¢¨¥¬. ’®£¤  a0 , a − 1, . . . , an ï¥âáï ¥£® ª®à­ï¬¨.

â® ¯à®â¨¢®à¥ç¨â ⥮६¥ 8.31. ‘«¥¤á⢨¥

8.35. ãáâì

­ ª®¢ãî äã­ªæ¨î ­  ’¥®à¥¬ 

R,

R { ¡¥áª®­¥ç­®¥ ¯®«¥. …᫨ ¤¢  ¬­®£®ç«¥­  f, g f (c) = g(c) ¤«ï ¢á¥å c ∈ R, â® f = g .

§ ¤ îâ ®¤¨-

â. ¥.

8.36. ãáâì

{ ¯®«¥ ¨§

R

§ ¤ ¥â ­ã«¥¢ãî äã­ªæ¨î ­ 

q

í«¥¬¥­â®¢. ’®£¤  ­¥­ã«¥¢®© ¬­®£®ç«¥­

Xq − X

R.

R∗ ¯®«ï R ®¡à §ãîâ ¬ã«ì⨯«¨ª â¨¢­ãî c ∈ R∗ , â® ¯®à冷ª c ¢ í⮩ £à㯯¥ ¤¥«¨â q−1 ¯® á«¥¤á⢨î 6.40. ®í⮬㠯® ¯à¥¤«®¦¥­¨î 6.16 cq−1 = 1, ®âªã¤  cq = c. â® à ¢¥­á⢮ ¢¥à­® ¨ ¤«ï ­ã«¥¢®£® „®ª § â¥«ìá⢮.

£à㯯㠯®à浪 

q−1.

‚ᥠ­¥­ã«¥¢ë¥ í«¥¬¥­âë

ɇǬ

í«¥¬¥­â . 5. Š®à­¨ ¬­®£®ç«¥­®¢ ­ ¤ ’¥®à¥¬ 

8.37 (Žá­®¢­ ï ⥮६   «£¥¡àë) . ãáâì

­ãî á⥯¥­ì. ’®£¤ 

f

C

¨

R

f ∈ C[X]

¨¬¥¥â ¯®«®¦¨â¥«ì-

¨¬¥¥â ª®¬¯«¥ªá­ë© ª®à¥­ì.

„®ª § â¥«ìá⢮.

à¨¢¥¤¥¬ ¤¢  ¤®ª § â¥«ìá⢠ í⮩ ⥮६ë.

¥à¢®¥ ¤®ª § â¥«ìá⢮

ãáâì

f

­¥ ¨¬¥¥â ª®¬¯«¥ªá­ëå ª®à­¥©. ’®£¤ 

⥮६¥ ‹¨ã¢¨««ï ®­  ¯®áâ®ï­­ . ® ⮣¤  ¨

f

F =

1 f

 ­ «¨â¨ç­  ¢

C ¨ ®£à ­¨ç¥­ .

®

¯®áâ®ï­­ , çâ® ­¥¢¥à­® ¯® á«¥¤á⢨î 8.35.

5. ŠŽˆ ŒŽƒŽ—‹…Ž‚ €„

C

ˆ

45

R

‚â®à®¥ ¤®ª § â¥«ìá⢮

8.38. ãáâì

‹¥¬¬ 

¨¬¥¥â ¯®«®¦¨â¥«ì­ãî á⥯¥­ì.

f ∈ C[X]

ɇǬ

z → ∞,

¯®

|f (z)| → ∞. „®ª § â¥«ìá⢮.

ɇǬ

f

¨¬¥¥â ¢¨¤

f = a0 + a1 X + · · · + an X n ,

an 6= 0.

(47)

’®£¤ 

  a0 |an−1 | |a0 | an−1 n |f (z)| = |z| |an + + · · · + n | ≥ |z| |an | − − ··· − n . |z| |z| |z| |z| n

‘ à®á⮬

|z|

ç¨á«®



®£à ­¨ç¥­® á­¨§ã ­¥ª®â®àë¬ ‹¥¬¬ 

8.39 (‹¥¬¬  „ « ¬¡¥à ) . …᫨

®ªà¥áâ­®áâ¨

z0

­ ©¤¥âáï â ª®¥ ç¨á«®

„®ª § â¥«ìá⢮.

¦¥­¨¥ ­ 

ãáâì

 |an−1 | |a0 | − ··· − n . |z| |z| M . ®í⮬ã |f (z)| ≥ |z|n M . |an | −

f (z0 ) 6= 0,

z1 ∈ C

¨

 §«®¦¨¬

f (z)

z,

f

çâ®

¨¬¥¥â ¢¨¤ (47) ¨

f (z0 ) 6= 0,

â® ¢ «î¡®©

|f (z)| < |f (z0 )|.

¢ àï¤ ’¥©«®à  ¢ ®ªà¥áâ­®áâ¨

z0 .

„¥«ï íâ® à §«®-

¯®«ãç ¥¬

f (z) = 1 + cp z p + cp+1 z p+1 + · · · + cn z n , cp , cn 6= 0. f (z0 ) 1 z1p = − . …᫨ z = z0 + tz1 , t ∈ (0, 1) ⊆ R, â® cp f (z) = 1 − tp + tp+1 hn−p−1 (t), f (z0 )

£¤¥

hn−p−1 (t) ∈ C[t]

¨¬¥¥â á⥯¥­ì

1 t< C(n − p)

n − p − 1.

ɇǬ

C

{ ¬ ªá¨¬ã¬ ¬®¤ã«¥© ª®íää¨æ¨¥­â®¢

hn−p−1 (t), â® ¯à¨ f (z) p p p+1 = 1 − tp (1 − C(n − p)t) < 1. f (z0 ) ≤ 1 − t + |hn−p−1 (t)| ≤ 1 − t + C(n − p)t f ∈ C[X] ¨¬¥¥â ¯®«®¦¨â¥«ì­ãî áâ¥∞ > M = inf z∈C |f (z)| > 0. ‚롥६ ¯®á«¥¤®¢ â¥«ì­®áâì â ª¨å zk , çâ® f (zk ) → M . …᫨ íâ  ¯®á«¥¤®¢ â¥«ì­®áâì ­¥®£à ­¨ç¥­ , â® ¨§ ¯®¤¯®á«¥¤®¢ â¥«ì­®áâì zik → ∞. ®«ãç ¥âáï ¯à®â¨¢®à¥ç¨¥ á «¥¬¬®©

‡ ¢¥à訬 ¢â®à®¥ ¤®ª § â¥«ìá⢮ ⥮६ë. …᫨ ¯¥­ì, ¨ ã

f

­¥â ª®à­¥©, â®

ª®¬¯«¥ªá­ëå ç¨á¥« ­¥¥ ¬®¦­® ¢ë¡à âì 8.38.

ãáâì ¯®á«¥¤®¢ â¥«ì­®áâì

zk

®£à ­¨ç¥­ . ‚ ᨫ㠯®«­®âë

C

¢ ­¥© ¬®¦­® ¢ë¡à âì

á室ïéãîáï ¯®¤¯®á«¥¤®¢ â¥«ì­®áâì. ¥§ ®£à ­¨ç¥­¨ï ®¡é­®á⨠¬®¦­® áç¨â âì, çâ® íâ® ’®£¤ 

f (zk ) → f (z0 ) = M .

‚ ᨫ㠫¥¬¬ë

8.40. ¥¯à¨¢®¤¨¬ë¥ ¬­®£®ç«¥­ë ­ ¤

C ¨¬¥îâ á⥯¥­ì 1.

‚ ç áâ­®áâ¨,

¨á室­ ï ¯®á«¥¤®¢ â¥«ì­®áâì. ˆâ ª,

zk → z0 .

„ « ¬¡¥à  ¯®«ãç ¥¬ ¯à®â¨¢®à¥ç¨¥. ‘«¥¤á⢨¥

ª ¦¤ë© ¬­®£®ç«¥­

f ∈ C[X]

®¤­®§­ ç­® ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥

f = a(X − c1 )k1 · · · (X − cm )km , £¤¥

c1 , . . . , cm

à §«¨ç­ë¥ ª®à­¨

„®ª § â¥«ìá⢮.

f

ªà â­®á⥩

a ∈ C,

k1 , . . . , km .

‚®á¯®«ì§®¢ âìáï ⥮६ ¬¨ 8.37, 8.28.

46

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

8.41. ãáâì

à¥¤«®¦¥­¨¥

„®ª § â¥«ìá⢮.

ãáâì

f

f ∈ R[X]

¨

c∈C

{ ª®à¥­ì

f.

’®£¤ 

c

â ª¦¥ ª®à¥­ì

f.

¨¬¥¥â ¢¨¤ 47. ’®£¤ 

0 = f (c) = a0 + a1 c + · · · + an cn = a0 + a1 c + · · · + an cn = a0 + a1 c + · · · + an cn = f (c).

“¯à ¦­¥­¨¥

ª®à¥­ì

f

8.42. ãáâì

ªà â­®áâ¨

’¥®à¥¬ 

f ∈ R[X]

¨

c∈C

{ ª®à¥­ì

8.43. ¥¯à¨¢®¤¨¬ë¥ ¬­®£®ç«¥­ë ­ ¤

ª ¦¤ë© ¬­®£®ç«¥­

f

ªà â­®áâ¨

k.

’®£¤ 

c

â ª¦¥

k. f ∈ R[X]

R

¨¬¥îâ á⥯¥­ì

≤ 2.

‚ ç áâ­®áâ¨,

®¤­®§­ ç­® ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥

f = a(X − c1 )k1 · · · (X − cm )km (X 2 + p1 X + q1 )l1 · · · (X 2 + ps X + qs )ls , £¤¥ c1 , . . . , cm à §«¨ç­ë¥ ª®à­¨ f ªà â­®á⥩ k1 , . . . R[X] à §«¨ç­ë ¨ ­¥ ¨¬¥îâ ¢¥é¥á⢥­­ëå ª®à­¥©.

, km ,

  ¬­®£®ç«¥­ë

a ∈ R,

2

(X + pi X + qi )li ∈

à¨¢¥¤¥¬ ®æ¥­ªã ¬®¤ã«¥© ª®à­¥© ª®¬¯«¥ªá­®£® ¬­®£®ç«¥­ . ’¥®à¥¬ 

ak+1 = 0,

¨

8.44. ãáâì

an , ak 6= 0,

£¤¥

f ∈ C[X] ¨¬¥¥â ¢¨¤ 0 ≤ k < n. ®«®¦¨¬

(47). à¥¤¯®«®¦¨¬, çâ®

an−1 = · · · =

B = max(|ak |, |ak−1 |, . . . , |a0 |). …᫨

z∈C

{ ª®à¥­ì

f,

„®ª § â¥«ìá⢮.

â®

B . |an |

s

B . |an |

|z| < 1 +

|z| ≥ 1 +

n−k

ãáâì

„®áâ â®ç­® ¯®ª § âì, çâ® n

s

n−k

|f (z)| > 0.

„¥©á⢨⥫쭮,

k

|f (z)| = |an z + ak z + · · · + a1 z + a0 | ≥ |an z n | − |ak z k + · · · + a1 z + a0 | ≥ Pk Pk |z|k+1 − 1 |an ||z|n − j=0 |aj ||z|j ≥ |an ||z|n − B j=0 |z|j = |an ||z|n − B = |z| − 1     |an | B B |an | B |z|n (|z| − 1) − |z|k+1 + > |z|n (|z| − 1) − |z|k+1 = |z| − 1 |an | |an | |z| − 1 |an |   k+1 |an ||z| B |z|n−k−1 (|z| − 1) − . |z| − 1 |an | ’ ª¨¬ ®¡à §®¬, ®áâ ¥âáï ¯®ª § âì, çâ®

|z|n−k−1 (|z| − 1) >

B . |an |

® ãá«®¢¨î

|z| − 1 ≥ α =

s

n−k

B > 0. |an |

®í⮬ã

|z|n−k−1 (|z| − 1) − ¨¡®

1 + α > α.

B ≥ (1 + α)n−k−1 α − αn−k = α[(1 + α)n−k−1 − αn−k−1 ] > 0, |an |

5. ŠŽˆ ŒŽƒŽ—‹…Ž‚ €„

C

ˆ

47

R

à¨¢¥¤¥¬  «£®à¨â¬ ˜âãଠ ¯à¨¡«¨¦¥­­®£® ­ å®¦¤¥­¨ï ¢¥é¥á⢥­­ëå ª®à­¥© ¬­®-

f ∈ R[X], â® ¯® á«¥¤áâf ¢¨î 8.25 ¬­®£®ç«¥­ ∈ R[X] ¨¬¥¥â ⥠¦¥ ª®à­¨, çâ® ¨ f , ­® ¨å ªà â­®á⨠áâ ­®¢ïâáï (f, f 0 ) f à ¢­ë¬¨ 1, â. ¥. ­¥ ¨¬¥¥â ªà â­ëå ª®à­¥©. (f, f 0 )

£®ç«¥­®¢ ¢¥é¥á⢥­­ë¬¨ ª®íää¨æ¨¥­â ¬¨. ‡ ¬¥â¨¬, çâ® ¥á«¨

’¥®à¥¬ 

8.45 (˜âãà¬) . ãáâì

á«¥¤®¢ â¥«ì­®áâì ˜âãଠ

f ∈ R[X] ­¥ ¨¬¥¥â ªà â­ëå ª®à­¥©. ®áâந¬ ¯®f1 = f 0 . „ «¥¥ ª ¦¤®¥ ¤«ï «î¡®£® i ≥ 2 ç¥à¥§ fi ¤¥«¥­¨ï fi−2 ­  fi−1 , â. ¥.

f0 = f,

®¡®§­ ç¨¬ ®áâ â®ª á® §­ ª®¬ - ®â

¨

fi−2 = gi−1 fi−1 − fi . ãáâì

u 0,

â®

fi−1 (c) =

−fi+1 (c). „®ª § â¥«ìá⢮.

ªà â­ë© ª®à¥­ì

f,

…᫨ ¡ë

fi (c) = fi+1 (c) = 0,

â® ¯® (48)

f (c) = f 0 (c) = 0,

â. ¥.

c

{

çâ® ­¥¢®§¬®¦­® ¢ ᨫã (8.24) (‘¬. â ª¦¥ ã¯à ¦­¥­¨¥ 8.32).

‚â®à®¥ ã⢥ত¥­¨¥ ¢ë⥪ ¥â ¨§ (48). ‹¥¬¬ 

8.47. ãáâì

fi−1 (x), fi+1 (x)

fi (c) = 0,

£¤¥

¨¬¥îâ à §­ë¥ §­ ª¨ ¯à¨

„®ª § â¥«ìá⢮.

i > 0. ’®£¤  áãé¥áâ¢ã¥â x ∈ (c − ε, c + ε).

ã¦­® ¢®á¯®«ì§®¢ âìáï ­¥¯à¥à뢭®áâìî

fj

â ª®¥

¤«ï ¢á¥å

ε > 0,

çâ®

j.

∈ (u, v) ­¥ ï¥âáï ª®à­¥¬ f .  áᬮâਬ fj (c) 6= 0, â® ¢ ­¥ª®â®à®© ®ªà¥áâ­®á⨠c ¢á¥ fj (c) ¨¬¥îâ ¯®áâ®ï­­ë© §­ ª. ®í⮬ã W (x) ¯®áâ®ï­­® ¢ í⮩ ®ªà¥áâ­®á⨠c. …᫨ ¦¥ ­¥ª®â®à®¥ fj (c) = 0, j ≥ 1, â® ¢ ­¥ª®â®à®© ®ªà¥áâ­®á⨠c ¯® «¥¬¬¥ 8.47 ä㭪樨 fj−1 (x), fj+1 (x) ¨¬¥îâ ¯®áâ®ï­­ë© §­ ª, ¨ ¯®í⮬ã ç¨á«® ¯¥à¥¬¥­ §­ ª®¢ ¢ àï¤ã ‡ ¢¥à訬 ¤®ª § â¥«ìá⢮ ⥮६ë. ãáâì

¯®¢¥¤¥­¨¥ ä㭪樨

W (x)

¢ ®ªà¥áâ­®áâ¨

fj−1 (x),

c.

…᫨ ¢á¥

fj (x),

(50)

fj+1 (x)

¯®áâ®ï­­®. ¥àï ¯¥à¥á¥ç¥­¨¥ íâ¨å ®ªà¥áâ­®á⥩

¤«ï ¢á¥å j , ¯®«ãç ¥¬ ®ªà¥áâ­®áâì c, ¢ W (x) ¯®áâ®ï­­®. ãáâì f (c) = 0. ® «¥¬¬¥ 8.46 ¨¬¥¥¬ f1 (c) 6= 0. ãáâì f1 (c) > 0. ’®£¤  ¢ ­¥ª®â®à®© ®ªà¥áâ­®á⨠U â®çª¨ c äã­ªæ¨ï f (x) ¢®§à áâ ¥â. ãáâì f (x) < 0, f (y) > 0, ¯à¨ ª®â®à®©

x, y ∈ U,

(51)

x < c < y.

…᫨ fj (c) 6= 0, â® ¬®¦­® áç¨â âì, çâ® fj (x) ¨¬¥¥â ¯®áâ®ï­­ë© §­ ª ¢ U . …᫨ fj (c) = 0, j > 1, â® ¬®¦­® áç¨â âì, çâ® ¢ àï¤ã (50) ç¨á«® ¯¥à¥¬¥­ §­ ª®¢ ¯®áâ®ï­­®. ‘«¥¤®¢ â¥«ì­®, ¤«ï

W (a) ¯¥à¥¬¥­ §­ ª®¢ ¢ àï¤ã (49) ¯®«ãç ¥¬ W (x) − W (y) = 1, ¥á«¨ x, y ¨§ (51). f1 (c) < 0. ’®£¤  ¢ ­¥ª®â®à®© ®ªà¥áâ­®á⨠U â®çª¨ c äã­ªæ¨ï f (x) ã¡ë¢ ¥â. ãáâì f (x) > 0, f (y) < 0, ¯à¨

ç¨á« 

ãáâì

x, y ∈ U,

(52)

x < c < y.

…᫨ fj (c) 6= 0, â® ¬®¦­® áç¨â âì, çâ® fj (x) ¨¬¥¥â ¯®áâ®ï­­ë© §­ ª ¢ U . …᫨ fj (c) = 0, j > 1, â® ¬®¦­® áç¨â âì, çâ® ¢ àï¤ã (50) ç¨á«® ¯¥à¥¬¥­ §­ ª®¢ ¯®áâ®ï­­®. ‘«¥¤®¢ â¥«ì­®, ¤«ï ç¨á« 

W (a)

¯¥à¥¬¥­ §­ ª®¢ ¢ àï¤ã (49) ¯®«ãç ¥¬

W (x) − W (y) = 1,

¥á«¨

x, y

¨§ (52).

48

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

6. ¥¯à¨¢®¤¨¬ë¥ ¬­®£®ç«¥­ë ­ ¤ Ž¯à¥¤¥«¥­¨¥

8.48. Œ­®£®ç«¥­ ¨§

Z[X] ­ §ë¢ ¥âáï

Z

¨

Q

¯à¨¬¨â¨¢­ë¬ , ¥á«¨ ¯à¨ç¥¬ ­ ¨-

¡®«ì訩 ®¡é¨© ¤¥«¨â¥«ì ¢á¥å ¥£® ª®íää¨æ¨¥­â®¢ à ¢¥­ 1. ’¥®à¥¬ 

8.49 (ƒ ãáá) . à®¨§¢¥¤¥­¨¥ ¯à¨¬¨â¨¢­ëå ¬­®£®ç«¥­®¢ ï¥âáï

¯à¨¬¨â¨¢­ë¬. „®ª § â¥«ìá⢮.

f, g ∈ Z[X] ¯à¨¬¨â¨¢­ë, ­® f g ­¥ ¯à¨¬¨â¨¢­®. ’®£¤  áãp, ¤¥«ï饥 ¢á¥ ª®íää¨æ¨¥­âë f g . ¥à¥©¤¥¬ ª ª®«ìæã ¢ëç¥â®¢ f, g ¢ Zp [X], â® ¢ Zp [X] ¯®«ãç ¥¬ f g = 0, çâ® ­¥¢®§¬®¦­®, â ª ª ª ãáâì

é¥áâ¢ã¥â ¯à®á⮥ ç¨á«®

Zp . …᫨ f , g { ®¡à §ë Zp [X] { ®¡« áâì. …᫨

f ∈ Q[X] \ 0,

â®

f =

¯à¨¬¨â¨¢­ë© ¬­®£®ç«¥­. ’¥®à¥¬ 

£¤¥

8.50. ãáâì

u, v ∈ Z[X],

¨

f ∈ Z[X] u = ±1e g , v = ±1e h.

„®ª § â¥«ìá⢮.

c, d ∈ Z,

¨

n e f, m

(c, d) = 1.

£¤¥

¯à¨¬¨â¨¢¥­, ¨

r g = ge, s rc e f= geh, ¨«¨ sd

¯à¨ç¥¬

f = gh,

r, s ∈ Z,

ˆ¬¥¥¬

ˆâ ª,

n, m ∈ Z,

¨

(n, m) = 1,

£¤¥

g, h ∈ Q[X].

(r, s) = 1.

fe ∈ Z[X],

¨

’®£¤ 

€­ «®£¨ç­®,

f = uv ,

h =

® ⥮६¥ 8.49 ­ ¨¡®«ì訩 ®¡é¨© ¤¥«¨â¥«ì ª®íää¨æ¨¥­â®¢ ¯à ¢®© ç á⨠(53) à ¢¥­

‘«¥¤á⢨¥

8.51. Œ­®£®ç«¥­

sd.

Žâá

f ∈ Z[X]

rs = ±cd.

­¥à §«®¦¨¬ ¢

Žâá

’¥®à¥¬ 

8.52 (Šà¨â¥à¨© ©§¥­è⥩­ ) . ãáâì

p - an ; p|an−1 , . . . , p|a1 , p|a0 ; (3) p2 - a0 . ’®£¤  ¬­®£®ç«¥­ f ­¥¯à¨¢®¤¨¬

p,

rs,

f = ±e ge h.

¢ ¯à®¨§¢¥¤¥­¨¥ ¬­®£®ç«¥-

Z[X]

­®¢ ¬¥­ì襩 á⥯¥­¨ ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤  ®­ ­¥¯à¨¢®¤¨¬ ¢

áãé¥áâ¢ã¥â â ª®¥ ¯à®á⮥ ç¨á«®

ce h, d (53)

(sd)f = (rc)e ge h,

  á ¤à㣮© áâ®à®­ë, ¯® (53) ®­ à ¢¥­

{

f ∈ Z[X]

Q[X].

¨¬¥¥â ¢¨¤ (47), ¯à¨ç¥¬

çâ®

(1) (2)

¢

Q[X].

d = ¢á¥å ª®íää¨æ¨¥­â®¢ f . ’®£¤  p - d. ®í⮬ã f ¬®¦­® f ¯à¨¬¨â¨¢¥­. ãáâì f = gh { ­¥âਢ¨ «ì­®¥ à §«®¦¥­¨¥ f ¢ Z[X] (á¬. ⥮६ã 8.50). ” ªâ®à¨§ãï ¯® ¬®¤ã«î p ¯®«ãç ¥¬ ¢ Zp [X] à §«®¦¥­¨¥ a0 X n = gh, £¤¥ ç¥àâ  á¢¥àåã ®§­ ç ¥â ®¡à § ¢ Zp [X] ¯® ¬®¤ã«î p. ‚ ᨫã ⥮६ë 8.22 „®ª § â¥«ìá⢮.

§ ¬¥­¨âì ­ 

d−1 f

ãáâì

¨ áç¨â âì, çâ®

¯®«ãç ¥¬, çâ®

g = bX r , £¤¥

h = cX n−r ,

b, c ∈ Z, ¨ 0 < r < n. ® ⮣¤  ᢮¡®¤­ë¥ ç«¥­ë g, h ¤¥«ïâáï ­  p, ¨ ¯®í⮬ã ᢮¡®¤­ë© f = gh ¤¥«¨âáï ­  p2 , çâ® ¯à®â¨¢®à¥ç¨â ¯à¥¤¯®«®¦¥­¨î.

ç«¥­

à¨¬¥à

Z[X]

8.53. ãáâì

p

{ ¯à®á⮥ ç¨á«®. ’®£¤  ¤«ï «î¡®£®

n≥1

¬­®£®ç«¥­

Xn + p ∈

­¥¯à¨¢®¤¨¬.

‘«¥¤á⢨¥

8.54. ãáâì

p

{ ¯à®á⮥ ç¨á«®. ’®£¤  横«®â®¬¨ç¥áª¨© ¬­®£®ç«¥­

Φp (X) = X p−1 + xp−2 + · · · + X + 1 = ­¥¯à¨¢®¤¨¬ ­ ¤

Q.

Xp − 1 X −1

7. €–ˆŽ€‹œ›… „Žˆ

®«®¦¨¬

„®ª § â¥«ìá⢮.

T = X + 1.

49

’®£¤ 

  p k T + · · · + pT T + pX + ··· + (T + 1)p − 1 k Φp (X) = = = T T   p k−1 T p−1 + pX p−2 + · · · + T + · · · + p. k p

p−1

Žáâ ¥âáï ¢®á¯®«ì§®¢ âìáï ªà¨â¥à¨© ©§¥­è⥩­ . 7.  æ¨®­ «ì­ë¥ ¤à®¡¨

a , b

ãáâì £¤¥

K { ª®¬¬ãâ â¨¢­ ï a, b ∈ K , ¯à¨ç¥¬ b 6= 0.

8.55. ‘ª ¦¥¬, çâ® ¤¢  ®â­®è¥­¨ï

Ž¯à¥¤¥«¥­¨¥ à¥¤«®¦¥­¨¥ “¯à ¦­¥­¨¥

F

a c , b d

à ¢­ë, ¥á«¨

ad = bc.

8.56. â® ®â­®è¥­¨¥ ï¥âáï ®â­®è¥­¨¥¬ íª¢¨¢ «¥­â­®áâ¨.

ab b = . ac c

8.57. ‘«¥¤ãî騥 ¤à®¡¨ à ¢­ë

Ž¡®§­ ç¥­¨¥

‚¢¥¤¥¬ ¢

®¡« áâì 楫®áâ­®áâ¨.  áᬮâਬ ¬­®¦¥á⢮ ®â­®è¥­¨©

8.58. —¥à¥§

F

®¡®§­ ç¨¬ ¬­®¦¥á⢮ ª« áᮢ à ¢­ëå ¤à®¡¥©.

á«®¦¥­¨¥ ¨ 㬭®¦¥­¨¥ ¯® ¯à ¢¨«ã

a c ad + bc + = , b d bd à¥¤«®¦¥­¨¥

ac ac = . bd bd

8.59. Ž¯à¥¤¥«¥­¨¥ á«®¦¥­¨ï ¨ 㬭®¦¥­¨ï ª®à४⭮.

F

®â­®á¨-

a . 1

’®£¤ 

⥫쭮 íâ¨å ®¯¥à æ¨ï ï¥âáï ¯®«¥¬. à¥¤«®¦¥­¨¥

¨­ê¥ªâ¨¢­®, ¨

8.60. ‡ ¤ ¤¨¬ ®â®¡à ¦¥­¨¥

8.61. ɇǬ

Ž¯à¥¤¥«¥­¨¥

8.62. ɇǬ

«¥¬¥­â

¢®¤¨¬ë© ¬­®£®ç«¥­, ¨ ’¥®à¥¬ 

φ

â®

F = Q.

K = R[X],

R.

«¥¬¥­â

£¤¥ R { ¯®«¥, â® F = R(X) ­ §ë¢ ¥âáï ¯®«¥¬ f ∈ R(X) ­ §ë¢ ¥âáï ¯à ¢¨«ì­®© ¤à®¡ìî , ¥á«¨ g

f ∈ R(X) ­ §ë¢ ¥âáï pk deg f < deg p.

¯à®á⥩襩 ¤à®¡ìî , ¥á«¨

p ∈ R[X]

{ ­¥¯à¨-

8.63. ‹î¡ ï ¤à®¡ì ï¥âáï á㬬®© ¬­®£®ç«¥­  ¨ ¯à®á⥩è¨å ¤à®¡¥©.

„®ª § â¥«ìá⢮. ‹¥¬¬ 

φ(a) =

P ⊆ F.

K = Z,

à æ¨®­ «ì­ëå ä㭪権 ­ ¤

deg f < deg g.

¯®« £ ï

φ(a + b) = φ(a) + φ(b), φ(ab) = φ(a)φ(b).

ˆâ ª, ¬®¦­® áç¨â âì, çâ® à¨¬¥à

φ : K → F,

 ¬ ¯®âॡã¥âáï

8.64. ãáâì

f ∈ R(X), g

¨

g = uv ,

£¤¥

u, v ∈ R[X],

¯à¨ç¥¬

(u, v) = 1.

’®£¤ 

f h t = + . g u v „®ª § â¥«ìá⢮.

vbf ,

ˆ¬¥¥¬

ua + vb = 1

¤«ï ­¥ª®â®àëå

¨

f uaf vbf af bf = + = + . g uv uv v u

a, b ∈ R[X].

Žâá

f = uaf +

50

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

f , pk deg f .

‚ ᨫã ⥮६ë 8.22 ¨ ¯à¥¤ë¤ã饩 «¥¬¬ë ¬®¦­® áç¨â âì, çâ® ¤à®¡ì ¨¬¥¥â ¢¨¤ £¤¥ ­ 

{ ­¥¯à¨¢®¤¨¬ë© ¬­®£®ç«¥­. ®ª ¦¥¬ ã⢥ত¥­¨¥ â¥®à¥¬ë ¨­¤ãªæ¨¥© ¯®

p

ɇǬ

deg f < deg p,

â® íâ  ¤à®¡ì ¯à®á⥩è ï. ãáâì

deg f ≥ deg p.

 §¤¥«¨¬

f

á ®áâ âª®¬

p, f = qp + r,

’®£¤ 

r=0

¨«¨

deg r < deg p.

f qp + r q r = = k−1 + k . pk pk p p

‚â®à®¥ á« £ ¥¬®¥ ï¥âáï ¯à ¢¨«ì­®© ¤à®¡ìî,   ¢ ¯¥à¢®¬ á⥯¥­ì ç¨á«¨â¥«ï 㬥­ì訫 áì. 8.65. Š ¦¤ ï à æ¨®­ «ì­ ï ¤à®¡ì ¨§

‘«¥¤á⢨¥

¬­®£®ç«¥­  ¨ ¤à®¡¥© ¢¨¤ 

c . (X − a)k

Š ¦¤ ï

C(X) ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥ á㬬ë à æ¨®­ «ì­ ï ¤à®¡ì ¨§ R(X) ¯à¥¤áâ ¢«ï-

¥âáï ¢ ¢¨¤¥ áã¬¬ë ¬­®£®ç«¥­  ¨ ¤à®¡¥© ¢¨¤ 

c , (X − a)k £¤¥

X 2 + pX + q ∈ R[X]

(X 2

aX + b , + pA + q)k

­¥ ¨¬¥¥â ¢¥é¥á⢥­­ëå ª®à­¥©. 8. Š®«ìæ® á⥯¥­­ëå à冷¢

ãáâì

R

{  áá®æ¨ â¨¢­®¥ ª®«ìæ® á 1.  áᬮâਬ ¬­®¦¥á⢮

R[[X]]

¢á¥å ¯®á«¥¤®¢ -

⥫쭮á⥩ (54)

a = (a0 , a1 , . . . ) í«¥¬¥­â®¢ ¨§ ¥á«¨

R.

Ž¯à¥¤¥«¨¬ ¢

R[[X]]

®¯¥à æ¨î á«®¦¥­¨ï ¯®-ª®®à¤¨­ â­®. Šà®¬¥ ⮣®,

¨§ (54),

a

b = (b0 , b1 , . . . ), â®

c = ab = (c0 , c1 , . . . ), £¤¥ ¤«ï «î¡®£®

k≥0 ck =

k X

(55)

ai bk−i .

i=0

à¥¤«®¦¥­¨¥

â â¨¢­®, â® ¨

Ž¯à¥¤¥«¥­¨¥ à¥¤«®¦¥­¨¥

¬®àä­®¥

R.

8.66.

R[[X]]

R[[X]]

8.67.

R[[X]]

R

ª®¬¬ã-

­ §ë¢ ¥âáï ª®«ì殬 á⥯¥­­ëå à冷¢ .

8.68. ‚á¥ í«¥¬¥­âë

(a0 , 0, . . . ) ∈ R[[X]] ®¡à §ãîâ ¯®¤ª®«ìæ®, ¨§®R[[X]] ®¡à §ãîâ ¯®¤ª®«ìæ®, ï¢-

‚ᥠ¯®ç⨠­ã«¥¢ë¥ ¯®á«¥¤®¢ â¥«ì­®á⨠¢

«ïî饥áï ª®«ì殬 ¬­®£®ç«¥­®¢ „®ª § â¥«ìá⢮. Ž¡®§­ ç¥­¨¥

R[X].

ï¥âáï  áá®æ¨ â¨¢­ë¬ ª®«ì殬 á 1. …᫨

ª®¬¬ãâ â¨¢­®.

®«®¦¨¬

R[X].

ˆ§®¬®à䨧¬ § ¤ ¥âáï ¯® ¯à ¢¨«ã

a 7→ (a, 0, . . . ) ∈ R[[X]].

8.69. ‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¬ë ¡ã¤¥¬ ®â®¦¤¥á⢫ïâì

a ∈ R á (a, 0, . . . ) ∈

X = (0, 1, 0, . . . ).

à¥¤«®¦¥­¨¥

8.70. „«ï «î¡®£®

n≥1

n

n

X = (0, . . . , 0, 1, 0, . . . ) ɇǬ

a

¨§ (54), â®

f = a0 + a1 X + · · · ,

ai ∈ R.

(56)

8. ŠŽ‹œ–Ž ‘’……›• Ÿ„Ž‚

a ∈ R[X] ¨§ (36), n. an ­ §ë¢ ¥âáï

8.71. ãáâì

Ž¯à¥¤¥«¥­¨¥

®à浪®¬

¯à¨ç¥¬

‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, à¥¤«®¦¥­¨¥

a0 = . . . = an−1 = 0, an 6= 0. f.

¬« ¤è¨¬ ç«¥­®¬

­ §ë¢ ¥âáï ç¨á«®

o(f )

51

R

{ ®¡« áâì.

8.72. ®à冷ª ¯à®¨§¢¥¤¥­¨ï à冷¢ à ¢¥­ á㬬¥ ¨å ¯®à浪®¢. Œ« ¤-

訩 ç«¥­ ¯à®¨§¢¥¤¥­¨ï à冷¢ à ¢¥­ ¯à®¨§¢¥¤¥­¨î ¬« ¤è¨å ç«¥­®¢ ᮬ­®¦¨â¥«¥©. ç áâ­®áâ¨,

8.73.

“¯à ¦­¥­¨¥

o(f + g) ≥ o(f ), o(g).

8.74. «¥¬¥­â

’¥®à¥¬ 

¢

‚

{ ®¡« áâì.

R[[X]]

¨§ (56) ®¡à â¨¬ ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤ 

f

a0

®¡à â¨¬

R. „®ª § â¥«ìá⢮.

ãáâì

g = b0 + b1 X + · · · = f −1 , ’®£¤ 

1 = gf ,

®âªã¤ 

Ž¡à â­®, ¯ãáâì

b0 = a−1 0 .

f

„«ï «î¡®£®

1 = a0 b0 .

€­ «®£¨ç­®, ¨§

¨§ (56) ¨

j≥1

a0

®¡à â¨¬® ¢

¨§ ãá«®¢¨ï

bi ∈ R.

(57)

1 = f g , ®âªã¤  1 = b0 a0 . R. ã¤¥¬ ¨áª âì g = f −1

¢ ¢¨¤¥ (57), £¤¥

¨¬¥¥¬

fg = 1

aj b0 + aj−1 b1 + · · · + a0 bj = 0. Žâá

bj = −a−1 0 [aj b0 + aj−1 b1 + · · · + a1 bj−1 ] . b j , j ≥ 1. g ®¡à â¨¬, â® f = f (gh) = (f g)h = h.

â  á¨á⥬  à ¢¥­á⢠¯®§¢®«ï¥â ¯®á«¥¤®¢ â¥«ì­® ®¯à¥¤¥«¨âì ¢á¥ ª®íää¨æ¨¥­âë ˆâ ª, ¤«ï ¤«ï

g

f

­ è¥«áï ¯à ¢ë© ®¡à â­ë© í«¥¬¥­â

­ ©¤¥âáï ¯à ¢ë© ®¡à â­ë©

h,

â. ¥.

g.

gh = 1.

’ ª ª ª ᢮¡®¤­ë© ç«¥­

Žâá

‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® ‘«¥¤á⢨¥

n = o(f ) ≥ 0

¨

u

8.75. Š ¦¤ë© ­¥­ã«¥¢®© í«¥¬¥­â

{ ®¡à â¨¬ë© í«¥¬¥­â ¨§

Ž¯à¥¤¥«¥­¨¥

¯®«¥ ¤à®¡¥©

à¥¤«®¦¥­¨¥

¥âáï ¢ ¢¨¤¥

8.76. ’ ª ª ª

R((X)).Ž­® X n u,

R[[X]]

f

R

{ ¯®«¥.

¨§

R[[X]]

f = X n u,

£¤¥

R[[X]].

{ ª®¬¬ãâ â¨¢­ ï ®¡« áâì, â® ¤«ï ­¥¥ áãé¥áâ¢ã¥â

­ §ë¢ ¥âáï ¯®«¥¬ «®à ­®¢áª¨å à冷¢ .

8.77. Š ¦¤ë© ­¥­ã«¥¢®© í«¥¬¥­â ¨§

£¤¥

¨¬¥¥â ¢¨¤

n∈Z

¨

u

{ ®¡à â¨¬ë© í«¥¬¥­â ¨§

R((X)) ®¤­®§­ ç­® ¯à¥¤áâ ¢«ïR[[X]].

ãáâì a = X m u, b = X m v , £¤¥ n, m ≥ 0, ¨ u, v { ®¡à â¨¬ë¥ í«¥a = X m−n uv −1 ¨¬¥¥â 㪠§ ­­ë© ¢¨¤. ¬¥­âë ¨§ R[[X]]. ’®£¤  b …᫨ X s u = X l v ∈ R((X)), ¨ s ≥ l, â® X s−l = uv −1 . ‘à ¢­¨¢ ï ᢮¡®¤­ë¥ ç«¥­ë, ¯®«ãç ¥¬ s − l = 0. ⮣¤  u = v . „®ª § â¥«ìá⢮.

‘«¥¤á⢨¥

8.78. ɇǬ

f ∈ R((X)) \ 0,

â®

f

®¤­®§­ ç­® ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥ «®à -

­®¢áª®£® à鸞

f = an X n + an+1 X n+1 + · · · , Ž¯à¥¤¥«¥­¨¥

8.79. ãáâì

f ∈ R((X))

n ∈ Z,

ai ∈ R,

an 6= 0.

(58)

¨§ (58). ®«®¦¨¬

f 0 = nan X n−1 + (n + 1)an+1 X n + · · · ∈ R((X)). à¥¤«®¦¥­¨¥

8.80. ɇǬ

„®ª § â¥«ìá⢮. ‘«¥¤á⢨¥

n 0

(g ) = ng

n−1 0

g

â®

0

0

(59) 0

0

(f + g) = f + g , (f g) = f g + f g

0

.

¥¯®á।á⢥­­ ï ¯à®¢¥àª  á ¨á¯®«ì§®¢ ­¨¥¬ (59) ¨ (55).

8.81. ɇǬ

¤«ï ¢á¥å

f, g ∈ R((X)),

0

f, g ∈ R((X)), n ∈ Z.

â®

(g −1 f )0 = g −2 (f 0 g − f g 0 ).

‚ ç áâ­®áâ¨,

52

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

XR[[X]] ¬­®¦¥á⢮ ¢á¥å g ∈ R[[X]], ¨¬¥îé¨å f ∈ R[[X]], g ∈ XR[[X]]. ®¤áâ ­®¢ª®© à鸞 g ¢ àï¤ f ¨§

8.82. Ž¡®§­ ç¨¬ ç¥à¥§

Ž¯à¥¤¥«¥­¨¥

¯®à冷ª ­¥ ¬¥­ìè¥ 1î ãáâì (56) ­ §®¢¥¬ àï¤

f (g) = a0 + a1 g + · · · + an g n + · · · 8.83.

Ž¯à¥¤¥«¥­¨¥

(60)

8.84. Ž¯à¥¤¥«¥­¨¥ ¯®¤áâ ­®¢ª¨ à鸞 ¢ àï¤

à¥¤«®¦¥­¨¥

ª®à४⭮. „®ª § â¥«ìá⢮.

ᮢ¯ ¤ ¥â á ª®íää¨æ¨¥­â®¬ ¯à¨ ’¥®à¥¬ 

8.85. ãáâì

„®ª § â¥«ìá⢮.

o(g n ) ≥ n, â® ¢ (60) ª®íää¨æ¨¥­â ¯à¨ X m a0 + a1 g + · · · + am−1 g m−1 .

„¥©á⢨⥫쭮, â ª ª ª

X

m

ã

f ∈ R[[X]], g ∈ XR[[X]]. Xm

Š®íää¨æ¨¥­â ¯à¨

ã

’®£¤ 

f (g)0

f (g)0 = f 0 (g)g 0 .

ᮢ¯ ¤ ¥â á ª®íää¨æ¨¥­â®¬ ¯à¨

m 0

m

X ã (a0 + a1 g + · · · + am g ) . ‘«¥¤®¢ â¥«ì­®, ¯® á«¥¤á⢨î 8.81 ª®íää¨æ¨¥­â f (g)0 à ¢¥­ a1 g 0 + · · · + mam g m−1 g 0 = (a1 + 2a2 g + · · · + mam g m−1 )g 0 .

¯à¨

Xm

ã

Žâá á«¥¤ã¥â ã⢥ত¥­¨¥. ‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® ¯®«¥

R

¨¬¥¥â ­ã«¥¢ãî å à ªâ¥à¨á-

⨪ã. à¥¤«®¦¥­¨¥ “¯à ¦­¥­¨¥

8.86. ãáâì

8.87. ãáâì

f, g ∈ R[[X]],

f ∈ R[[X]]

8.88. ãáâì

f 0 = g0 .

’¥®à¥¬ 

8.89. ɇǬ

f ∈ XR[[X]].

„®ª § â¥«ìá⢮.

i,j≥0

ln(1 + f ) =

X (−1)i−1 j≥1

i

exp(f + g) = exp f exp g ,

¨

f i.

(ln(1 + f ))0 =

f0 . 1+f

ˆ¬¥¥¬

f + ··· + 1! P fi fj 1P = t≥0 i! j! t! i+j=t

exp f exp g = (1 + P

â®

j≥0

®«®¦¨¬

f f + ··· + + ··· , 1! n! o(f ), o(g) ≥ 1,

f = g + c, c ∈ R.

f (j) (0) . j!

n

exp(f ) = 1 +

’®£¤ 

¨§ (56). ’®£¤  ¤«ï «î¡®£®

aj = Ž¯à¥¤¥«¥­¨¥

¯à¨ç¥¬

fn g gn + · · · )(1 + + · · · + + ···) = n! 1! n! t! i j P 1 f g = t≥0 (f + g)t = exp(f + g). i!j! t!

„«ï ¤®ª § â¥«ìá⢠ ¢â®à®£® ã⢥ত¥­¨ï § ¬¥â¨¬, çâ® ¯® ⥮६¥ 8.85

(ln(1 + f ))0 =

X

(−1)j−1 f i−1 f 0 = (1 + f )−1 f 0 .

j≥1

’¥®à¥¬ 

8.90. ɇǬ

o(f ) ≥ 1,

â®

ln(exp f ) = f, „®ª § â¥«ìá⢮.

exp(ln(1 + f )) = f.

„®áâ â®ç­® ¯®ª § âì ã⢥ত¥­¨¥ ¯à¨

f = X.

‚ëç¨á«¨¬ ¯à®¨§-

¢®¤­ãî, ¨á¯®«ì§ãï ⥮६ã 8.89,

(ln(exp X))0 = (ln(1 + (exp X − 1)))0 =

(exp X − 1)0 exp X = = 1 = X 0. 1 + (exp X − 1) exp X

8. ŠŽ‹œ–Ž ‘’……›• Ÿ„Ž‚

Žâá

ln(exp X) = X + c, c ∈ R ln(exp 0) = 0 = c.

53

¯® ¯à¥¤«®¦¥­¨î ¯à¥¤«®¦¥­¨î 8.86. ®¤áâ ¢«ïï

X = 0,

¯®«ãç ¥¬

 áᬮâਬ ⥯¥àì àï¤

exp(ln(1 + X)) 1+X ¨ ¢ëç¨á«¨¬ ¥£® ¯à®¨§¢®¤­ãî, ¨á¯®«ì§ãï á«¥¤á⢨¥ 8.81 ¨ ⥮६ã 8.89



exp(ln(1 + X)) 1+X

0

=

[exp(ln(1 + X))]0 (1 + X) − exp(ln(1 + X)) = (1 + X)2

exp(ln(1+X)) (1 1+X

+ X) − exp(ln(1 + X)) (1 + X)2

=0

Žâá

exp(ln(1 + X)) ∈ R. 1+X ®¤áâ ¢«ïï

X=0

¯®«ãç ¥¬

exp(ln(1 + X)) = exp(ln 1)) = exp 0 = 1. 1+X 8.91. Œ­®¦¥á⢮ XR[[X]] ï¥âáï £à㯯®© ®â­®á¨â¥«ì­® á«®¦¥­¨ï. 1 + XR[[X]] { ¬­®¦¥á⢮ ¢á¥å à冷¢ ¨§ R[[X]] ᮠ᢮¡®¤­ë¬ ç«¥­®¬ 1. ’®£¤  1 + XR[[X]] { £à㯯  ®â­®á¨â¥«ì­® 㬭®¦¥­¨ï. Žâ®¡à ¦¥­¨¥ exp : XR[[X]] → 1 + XR[[X]] § ¤ ¥â ¨§®¬®à䨧¬ íâ¨å £à㯯. Ž¡à â­®¥ ®â®¡à ¦¥­¨¥ ª ­¥¬ã ¨¬¥¥â ¢¨¤ ln : 1 + XR[[X]] → XR[[X]]. ‚ ç áâ­®áâ¨, ln((1 + f )(1 + g)) = ln(1 + f ) + ln(1 + g) ¤«ï ¢á¥å f, g ∈ XR[[X]]. ‘«¥¤á⢨¥

ãáâì

54

8. ŒŽƒŽ—‹…› ˆ Ÿ„› Ž’ Ž„Ž‰ ……Œ…Ž‰

ƒ‹€‚€ 9

Œ­®£®ç«¥­ë ®â ­¥áª®«ìª¨å ¯¥à¥¬¥­­ëå

1. Š®«ìæ® ¬­®£®ç«¥­®¢ ®â ­¥áª®«ìª¨å ¯¥à¥¬¥­­ëå

9.1. ãáâì

Ž¯à¥¤¥«¥­¨¥

R

{  áá®æ¨ â¨¢­®¥ ª®«ìæ® á 1. ® ¨­¤ãªæ¨¨ ¯®«®¦¨¬

R[X, . . . , Xn ] = (R[X, . . . , Xn−1 ])[Xn ]. ’ ª¨¬ ®¡à §®¬, ª ¦¤ë© í«¥¬¥­â ¨§

R[X, . . . , Xn ] ®¤­®§­ ç­® ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥ á㬬ë

®¤­®ç«¥­®¢

ai1 ,... ,in X1i1 · · · Xnin ,

ai1 ,... ,in ∈ R,

(61)

, in ∈ N∪0. ãáâì (N∪0)n { ¬­®¦¥á⢮ ¬ã«ì⨨­¤¥ªá®¢ , â. ¥. ¬­®¦¥á⢮ ­ ¡®à®¢ (i1 , . . . , in ) í«¥¬¥­â®¢ ¨§ N ∪ 0. ’®£¤  ®¤­®ç«¥­ (61) § ¯¨á뢠¥âáï ¢ ¢¨¤¥

£¤¥ i1 , . . .

ai X i ,

X i = X1i1 · · · Xnin .

i = (i1 , . . . , in ) ∈ (N ∪ 0)n ,

ˆ§ á«¥¤á⢨ï 8.8 ¢ë⥪ ¥â ’¥®à¥¬ 

9.2. ɇǬ

R

{ ®¡« áâì, â® ¨

„«ï áà ¢­¥­¨ï ®¤­®ç«¥­®¢ ¢¢¥¤¥¬ ¢

R[X, . . . , Xn ]

(N ∪ 0)n

{ ®¡« áâì.

®â­®è¥­¨¥ ¯®à浪 .

9.3. ãáâì

Ž¯à¥¤¥«¥­¨¥

m = (m1 , . . . , mn ), r = (r1 . . . , rn ) ∈ (N ∪ 0)n . ‘ª ¦¥¬, çâ®

m > r,

¥á«¨ áãé¥áâ¢ã¥â â ª®¥

1 ≤ j < n,

çâ® (62)

m1 = r1 , . . . , mj−1 = rj−1 , mj > rj . 9.4. …᫨

à¥¤«®¦¥­¨¥

0

0

m ≥ m , m ≥ m,

â®

m > m0 , m0 > m00 ,

â®

m > m00 .

Šà®¬¥ ⮣®,

0

m ≥ m,

¨ ¥á«¨

m=m.

„®ª § â¥«ìá⢮.

ãáâì

m = (m1 , . . . , mn ), m0 = (m01 . . . , m0n ), m00 = (m001 . . . , m00n ).

(63)

ãáâì

m1 = m01 , . . . , mj−1 = m0j−1 , mj > m0j ;

(64)

m01 = m001 , . . . , m0j 0 −1 = m00j 0 −1 , m0j 0 > m00j 0 . …᫨

s = min(j, j 0 ),

â®

m1 = m001 , . . . , ms−1 = m00s−1 , ms > m00s . ®í⮬ã

m > m00 .

’¥®à¥¬ 

€­ «®£¨ç­® ¤®ª §ë¢ îâáï ®áâ «ì­ë¥ ã⢥ত¥­¨ï.

9.5. ‹î¡ ï ¯®á«¥¤®¢ â¥«ì­®áâì

¥âáï. 55

M1 ≥ M2 ≥ . . .

¢

(N ∪ 0)n

áâ ¡¨«¨§¨àã-

56

9. ŒŽƒŽ—‹…› Ž’ …‘ŠŽ‹œŠˆ• ……Œ…›•

„®ª § â¥«ìá⢮.

„®ª § â¥«ìá⢮ ¯à®¢®¤¨âáï ¨­¤ãªæ¨¥© ¯®

ã⢥ত¥­¨¥ ®ç¥¢¨¤­®. ãáâì ¤«ï â® ¯® ãá«®¢¨î

....

m11 ≥ m21 ≥ . . . .

n−1

‘«¥¤®¢ â¥«ì­®, áãé¥áâ¢ã¥â

(N ∪ 0)n−1

® ãá«®¢¨î ¢ í⮬ á«ãç ¥ ¢

n. …᫨ n = 1, â® Mi = (mi1 , mi2 , . . . , min ), â ª®¥ k , çâ® mk,1 = mk+1,1 =

⥮६  ¤®ª § ­ . …᫨ ¯®«ãç ¥¬

(mk,2 , . . . , mk,n ) ≥ (mk+1,2 , . . . , mk+1,n ) ≥ . . . ® ¨­¤ãªæ¨¨ íâ  ¯®á«¥¤®¢ â¥«ì­®áâì áâ ¡¨«¨§¨àã¥âáï ­ ç¨­ ï á ­¥ª®â®à®£® ¬¥áâ 

t.

’®£¤ 

Mt = Mt+1 = . . . . 9.6. ɇǬ

à¥¤«®¦¥­¨¥

„®ª § â¥«ìá⢮.

m1 +

m001

‘«¥¤á⢨¥

0

m +m

000

=

m01

+

m, m0 , m00 ∈ (N ∪ 0)n

ãáâì

m001 , . . .

9.7. ɇǬ

m, m0 , m00

m > m0 ,

¨

â®

m + m00 > m0 + m00 .

¨§ (63) ¨ ¢ë¯®«­¥­® (64). ’®£¤ 

, mj−1 + m00j−1 = m0j−1 + m00j−1 , mj + m00j > m0j + m00j .

m, m0 , m00 , m000 ∈ (N ∪ 0)n

¨

m ≥ m0 , m00 ≥ m000 ,

â®

m + m00 >

.

Ž¯à¥¤¥«¥­¨¥

9.8. ãáâì

X

f=

ai X i 6= 0.

(65)

i∈(N∪0)n

am X m ¨§ í⮣® ¯à¥¤áâ ¢«¥­¨ï j ∈ (N ∪ 0)n , çâ® aj 6= 0.

Ž¤­®ç«¥­ â ª¨å

­ §ë¢ ¥âáï áâ à訬 ¢

‚áî¤ã ¢ ¤ «ì­¥©è¥¬ ¢ í⮩ £« ¢¥ ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ®

f,

¥á«¨

R

m>j

¤«ï ¢á¥å

{  áá®æ¨ â¨¢­®-

ª®¬¬ãâ â¨¢­ ï ®¡« áâì á 1. ˆ§ á«¥¤á⢨ï 9.7 ¢ë⥪ ¥â ’¥®à¥¬ 

9.9. ‘â à訩 ®¤­®ç«¥­ ¯à®¨§¢¥¤¥­¨ï ¬­®£®ç«¥­®¢ à ¢¥­ ¯à®¨§¢¥¤¥­¨î

áâ àè¨å ®¤­®ç«¥­®¢. Ž¯à¥¤¥«¥­¨¥

¢®¬ ®¤­®ç«¥­¥

“¯à ¦­¥­¨¥

f ¨§ (9.7) ®¤­®à®¤¥­ á⥯¥­¨ d, ¥á«¨ ¢ «î¡®¬ ­¥­ã«¥i = (i1 , . . . , in ), ¢ë¯®«­¥­® ãá«®¢¨¥ i1 + · · · + in = d.

9.10. Œ­®£®ç«¥­

ai X i

¨§ (9.7), £¤¥

9.11. ãáâì

f

¨§ (9.7) ¨

fd

{ ¥£® ®¤­®à®¤­ ï ª®¬¯®­¥­â  á⥯¥­¨

¥. á㬬  ¢á¥å ¥£® ®¤­®à®¤­ëå ®¤­®ç«¥­®¢ á⥯¥­¨

d.

’®£¤ 

d,

â.

f = f0 + f1 + · · · .

2. ‘¨¬¬¥âà¨ç­ë¥ ¬­®£®ç«¥­ë Ž¯à¥¤¥«¥­¨¥

9.12. Œ­®£®ç«¥­

f

¨§ (9.7) ᨬ¬¥âà¨ç¥­ , ¥á«¨ ®­ ­¥ ¬¥­ï¥âáï ¯à¨ «î-

¡®© ¯¥à¥áâ ­®¢ª¥ ¯¥à¥¬¥­­ëå. à¨¬¥à ¬¨ ᨬ¬¥âà¨ç­ëå ¬­®£®ç«¥­®¢ ïîâáï í«¥¬¥­-

â à­ë¥ ᨬ¬¥âà¨ç­ë¥ ¬­®£®ç«¥­ë

σ1 = σ1 (X1 , . . . , Xn ) = X1 + · · · + Xn ; σ2 = σ2 (X1 , . . . , Xn ) = X1 X2 + · · · + Xn−1 Xn ; ..................................................... σn = σn (X1 , . . . , Xn ) = X1 · · · Xn . „à㣨¬¨ á«®¢ ¬¨,

σk

{ íâ® á㬬  ¢á¥å ¯à®¨§¢¥¤¥­¨© ¯®

k

­¥¨§¢¥áâ­ëå

(66)

X i1 · · · X ik , 1 ≥ i 1


¨¬¥¥¬

pk − pk−1 σ1 + pk−2 σ2 + · · · + (−1)k−1 p1 σk−1 + (−1)k σk = 0. „®ª § â¥«ìá⢮.

‡ ¬¥â¨¬, çâ®

F (X) =

Z[X1 , . . . , Xn ] ⊆ Q[X, X1 , . . . , Xn ].

n X

σk X k =

k=0

n Y

® (70)

(1 + Xk X).

k=1

’®£¤  ¯® (8.91)

ln(F (X)) =

n X

ln(1 + Xk X).

k=1

‚ëç¨á«ïï ¯à®¨§¢®¤­ãî ¯® 0

F (X) = F (X)

¯® ⥮६¥ 8.89 ¯®«ãç ¥¬, çâ®

X

Pn Xk = k=1 Xk (1 − Xk X + Xk2 X 2 + · · · ) = 1 + Xk X p1 − Xp2 + X 2 p3 + · · · + (−1)k T k pk+1 + · · ·

Pn

k=1

‘ ¤à㣮© áâ®à®­ë,

F (X)0 = F (X) ®í⮬ã n X

σk kX k−1 = (

k=1

ˆâ ª, ¯à¨

n X

Pn k−1 k=1 σk kX P . n k k=1 σk X

σk X k )(p1 − Xp2 + X 2 p3 + · · · + (−1)k T k pk+1 + · · · )

k=1

k = 1, . . . , n kσk = (−1)k pk + (−1)k−1 σ1 + · · · + p1 σk−1 .

Žâá ­ å®¤¨¬ ‘«¥¤á⢨¥

pk 9.18. ɇǬ

R

{ ¯®«¥ ­ã«¥¢®© å à ªâ¥à¨á⨪¨, â® ª ¦¤ë© ᨬ¬¥âà¨ç-

­ë© ¬­®£®ç«¥­ ï¥âáï ¬­®£®ç«¥­®¬ ®â “ª ¦¥¬ ï¢­ë¥ ¢ëà ¦¥­¨ï

σk

ç¥à¥§

p1 , . . . , pn .

p1 , . . . , pn

¨ ­ ®¡®à®â. ‚ ᨫã ⥮६ë 9.17 ¨¬¥¥¬

á¨á⥬㠫¨­¥©­ëå ãà ¢­¥­¨©

 σ1 = p1     −2σ2 = p2  σ 1 p1 σ 1 p2 −σ2 p1 3σ2 = p3   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......    σ1 pk−1 −σ2 pk−2 σ3 pk−2 . . . (−1)k−1 kσk = pk

(72)

3. „ˆ‘ŠˆŒˆ€’ ˆ …‡“‹œ’€’

59

¥è ï á¨á⥬ã (72) ®â­®á¨â¥«ì­®

σk ¯® ¯à ¢¨«ã Šà ¬¥à , ¯®«ãç ¥¬ 1 0 0 . . . 0 p1 p1 −2 0 . . . 0 p2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p1 1 0 . . . 0 0 pk−1 pk−2 . . . . . . p1 pk 1 p2 p1 2 . . . 0 0 = σk = . k! . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 ... 0 0 1 pk pk−1 . . . . . . p2 p1 p1 −2 0 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pk−1 pk−2 . . . . . . p1 (−1)k−1 k

¥è ï á¨á⥬ã (72) ®â­®á¨â¥«ì­®

pk ¯® ¯à ¢¨«ã Šà ¬¥à , ¯®«ãç ¥¬, σ1 1 0 ... 0 0 2σ2 σ1 1 ... 0 0 pk = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (k − 1)σk−1 σk−2 . . . . . . σ1 1 kσk σk−1 . . . . . . σ2 σ − 1

‚ ç áâ­®áâ¨,

p2 = σ12 − 2σ2 ,

çâ®

(73)

p3 = σ13 − 3σ1 σ2 + 3σ3 .

3. „¨áªà¨¬¨­ ­â ¨ १ã«ìâ ­â

 áᬮâਬ ¢

Z[X1 , . . . , Xn ]

¬­®£®ç«¥­

W (X1 , . . . , Xn ) =

Y

(Xj − Xi ).

1≤i<j≤n

Œ­®£®ç«¥­

W (X1 , . . . , Xn )

X1 , . . . , Xn . ®í⮬ã Wn (X1 , . . . , Xn )2 Z[X1 , . . . , Xn ].  ©¤¥¬ ¥£® ¢ëà ¦¥­¨¥ ç¥à¥§

ª®á®á¨¬¬¥âà¨ç¥­ ¯®

ï¥âáï ᨬ¬¥âà¨ç¥áª¨¬ ¬­®£®ç«¥­®¬ ¢

σ1 , . . . , σn ¯® ä®à¬ã« ¬ (73). ‚ á ¬®¬ ¤¥«¥, 1 1 ... 1 1 X1 X12 . . . X1n−1 X1 X2 ... Xn 1 X2 X22 . . . X2n−1 X22 ... Xn2 W (X1 , . . . , Xn )2 = X12 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n−1 1 Xn X 2 . . . X n−1 n n X X2n−1 . . . Xnn−1 1 n p1 p2 . . . pn−1 p1 p2 p3 ... pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pn−1 pn pn+1 . . . p2n−2

p1 , . . . , pn

¨ ⥬ á ¬ë¬ ç¥à¥§

‚ ç áâ­®áâ¨, ¯à¨

n=2

2 Disc(x1 , x2 ) = p1

Ž¯à¥¤¥«¥­¨¥

9.19. ˆ¬¥¥¬

p1 = 2p2 − p21 = 2(σ12 − 2σ2 ) − σ12 = σ12 − 4σ2 . p2

W (X1 , . . . , Xn )2 = Disc(σ1 , . . . , σn ). Œ­®£®ç«¥­

Disc(X1 , . . . , Xn ) ∈ Z[X1 , . . . , Xn ] ¨§ (74) ­ §ë¢ ¥âáï ¤¨áªà¨¬¨­ ­â®¬ . ˆ§ í⮣® ®¯à¥¤¥«¥­¨ï ¢ë⥪ ¥â

(74)

60

9. ŒŽƒŽ—‹…› Ž’ …‘ŠŽ‹œŠˆ• ……Œ…›•

’¥®à¥¬ 

9.20. ãáâì

f = X n + an−1 X n−1 + · · · + a0 ∈ R[X], ,

R

{ ¯®«¥, ¨

α1 , . . . , αn

{ ¥£® ª®à­¨. ’®£¤ 

Disc(σ1 (α1 , . . . , αn ), . . . , σn (α1 , . . . , αn ))) = 0 ¢ ⮬ ¨ ⮫쪮 ¢ ⮬ á«ãç ¥, ¥á«¨

f

¨¬¥¥â ªà â­ë© ª®à¥­ì.

9.21. ãáâì § ¤ ­ ¬­®£®ç«¥­

Ž¯à¥¤¥«¥­¨¥

f = an X n + an−1 X n−1 + · · · + a0 , à¥¤¯®«®¦¨¬, çâ®

α1 , . . . , αn { ¥£® ª®à­¨. Disc(f ) =

an 6= 0.

„¨áªà¨¬¨­ ­â®¬ í⮣® ¬­®£®ç«¥­  ­ §ë¢ ¥âáï

a2n−2 W (α1 , . . . n

, αn )2 .

‚ ᨫã (9.19) ¨ ä®à¬ã« ‚¨¥â  ¯®«ãç ¥¬

Disc(f ) = a2n−2 Disc(− n ’¥¬ á ¬ë¬ ¬®¦­® ¢ëç¨á«¨âì

Disc(f )

a0 an−1 , . . . , (−1)n ). an an

ç¥à¥§ ª®íää¨æ¨¥­âë

an , . . . , an .

“ª ¦¥¬ ë©

ᯮᮡ í⮣® ¢ëç¨á«¥­¨ï. „«ï í⮣® ¢¢¥¤¥¬ ¯®­ï⨥ १ã«ìâ ­â  ¤¢ãå ¬­®£®ç«¥­®¢. ãáâì ¤ ­ë ¤¢  ¬­®£®ç«¥­ 

f = an X n + an−1 X n−1 + · · · + a0 ,

g = bm X m + bm−1 X m−1 + · · · + b0

á ª®íää¨æ¨¥­â ¬¨ ¢ ¯®«¥

R. à¨ í⮬ ¬ë ¯à¥¤¯®« £ ¥¬, çâ®, ¯® ªà ©­¥© ¬¥à¥, ®¤¨­ an , bm ®â«¨ç¥­ ®â ­ã«ï. ®«®¦¨¬   an an−1 an−2 . . . a0 0 0 ... 0  0 an an−1 . . . a1 a0 0 ... 0    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    0 0 0 . . . an an−1 . . . . . . . . a0    R(f, g) =  bm bm−1 bm−2 . . . b1 ∈ Mat(m + n, R). b0 0 ... 0     0  bm bm−1 . . . b2 b1 b0 . . . 0     .. .. .. .. .. .. .. . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . bm . . . . . . . . . . . . . . . . . . . . b0

¨§

ª®íää¨æ¨¥­â®¢

‚ í⮩ ¬ âà¨æ¥ ¯¥à¢ë¥ ç«¥­ 

f,

  ¯®á«¥¤­¨¥

à¥¤«®¦¥­¨¥

n áâப § ¯®«­¥­ë ᮠᤢ¨£®¬ ­  ®¤¨­ è £ ª®íää¨æ¨¥­â ¬¨ ¬­®£®m áâப { ª®íää¨æ¨¥­â ¬¨ ¬­®£®ç«¥­  f . ˆ§ ¢¨¤  R(f, g) ¢ë⥪ ¥â

9.22. ‘¯à ¢¥¤«¨¢® à ¢¥­á⢮

 m−1  X f m−2   X f  n+m−1   X  ..   . X n+m−2          f   . . R(f, g)   =  n−1  . g  X    X n−2 g   X    1  .   ..  g Ž¯à¥¤¥«¥­¨¥

9.23. ¥§ã«ì⠭⮬ ¬­®£®ç«¥­®¢

f, g

(75)

­ §ë¢ ¥âáï

R(f, g) = det(R(f, g)). ’¥®à¥¬ 

9.24. ‘ãé¥áâ¢ãîâ â ª¨¥ ¬­®£®ç«¥­ë

u, v ∈ R[X],

çâ®

R(f, g) = f u + gv.

3. „ˆ‘ŠˆŒˆ€’ ˆ …‡“‹œ’€’

„®ª § â¥«ìá⢮.

ɇǬ

R(f, g) = 0,

61

â® ã⢥ত¥­¨¥ ®ç¥¢¨¤­®. ãáâì

R(f, g) 6= 0.

¥è ï á¨á⥬ã (75) ®â­®á¨â¥«ì­® 1 ¯® ¯à ¢¨«ã Šà ¬¥à , ¯®«ãç ¥¬

Uf + V G , R(f, g) U V u= ,v = . R(f, g) R(f, g) 1=

Žáâ ¥âáï ¯®«®¦¨âì ’¥®à¥¬ 

(1) (2)

U, V ∈ R[X].

9.25. ‘«¥¤ãî騥 ãá«®¢¨ï íª¢¨¢ «¥­â­ë:

R(f, g) 6= 0; (f, g) = 1.

„®ª § â¥«ìá⢮.

ãáâì

(f, g) = 1

¨

R(f, g) = 0.

’®£¤  áâப¨

R(f, g)

«¨­¥©­®

§ ¢¨á¨¬ë. ®í⮬㠭 ©¤¥âáï â ª ï ­¥­ã«¥¢ ï áâப 

Λ = (λn+m−1 , . . . , λ0 ), çâ®

®

ΛR(f, g) = 0. ‚ ᨫã (75) ¯®«ãç ¥¬, çâ®  m−1  X f X m−2 f      . .   .    f  m−1  Λ  n−1  + · · · + λn )f + (λn−1 X n−1 + · · · + λ0 )g = 0.  = (λn+m−1 X  X n−2 g  X g     . .   . g (f, g) = 1.

®í⮬㠨§ ¯®á«¥¤­¥£® à ¢¥­á⢠ ¢ë⥪ ¥â, çâ®

f |(λn−1 X n−1 + · · · + λ0 ),

g|(λn+m−1 X m−1 + · · · + λn ).

â® ¢®§¬®¦­® «¨èì ¢ á«ãç ¥, ª®£¤ 

Λ = 0, â ª ª ª «¨¡® deg f = n, «¨¡® deg g = m. R(f, g) 6= 0. ® ⥮६¥ 9.24 áãé¥áâ¢ãîâ â ª¨¥ ¬­®£®ç«¥­ë u, v , u u + g R(f,g) , ¨ ¯®í⮬ã (f, g) = 1. R(f, g) = f u + gv . ’®£¤  1 = f R(f,g) Ž¡à â­®, ¯ãáâì

‘«¥¤á⢨¥

(1) (2)

R(f, g) = 0; f, g ¨¬¥îâ ®¡é¨©

‘«¥¤á⢨¥

â®

9.26. ãáâì

f, g ∈ C[X].

‘«¥¤ãî騥 ãá«®¢¨ï íª¢¨¢ «¥­â­ë:

ª®à¥­ì.

9.27. ɇǬ

R

{ ¯à®¨§¢®«ì­®¥ ¯®«¥, ¨

f, g ∈ R[X]

¨¬¥îâ ®¡é¨© ª®à¥­ì,

R(f, g) = 0.  ¬ ¯®âॡã¥âáï á«¥¤ãî饥 à¥¤«®¦¥­¨¥

9.28. ãáâì

R

{ ª®¬¬ãâ â¨¢­®- áá®æ¨ â¨¢­ ï ®¡« áâì á 1, ¨

f (X1 , . . . , Xn ) ∈ R[X1 , . . . , Xn ]. …᫨

f (X1 , X2 , X3 , . . . , Xn ) = 0,

â®

f (X1 , . . . , Xn ) = (X1 − X2 )h(X1 , . . . , Xn ), „®ª § â¥«ìá⢮.

çâ®

h(X1 , . . . , Xn ) ∈ R[X1 , . . . , Xn ].

‡ ¬¥â¨¬, çâ®

R[X1 , . . . , Xn ] = R[X1 , X1 − X2 , X3 , . . . , Xn ]. ®í⮬ã

f (X1 , . . . , Xn ) = u(X1 , X3 , . . . , Xn ) + (X1 − X2 )h(X1 , . . . , Xn ). Žâá á«¥¤ã¥â ã⢥ত¥­¨¥.

62

9. ŒŽƒŽ—‹…› Ž’ …‘ŠŽ‹œŠˆ• ……Œ…›•

’¥®à¥¬ 

9.29. ãáâì

{ ª®¬¬ãâ â¨¢­®- áá®æ¨ â¨¢­ ï ®¡« áâì á 1,

K

R = K[X1 , . . . , Xn , Y1 , . . . , Ym ]. ‚

R[X]

¯®«®¦¨¬

f = an X n + an−1 X n−1 + · · · + a0 , ak = (−1)k an σ(X1 , . . . , Xn ), g = bm X m + bm−1 X m−1 + · · · + b0 , bk = (−1)k bm σ(Y1 , . . . , Ym ), £¤¥

an , bn ∈ K

(76)

­¥ à ¢­ë ­ã«î. ’®£¤  n R(f, g) = am n bm

Y (Xi − Yj ). i,j

„®ª § â¥«ìá⢮.

Xi

¥§ ®£à ­¨ç¥­¨ï ®¡é­®á⨠¬®¦­® áç¨â âì, çâ®

®â®¦¤¥á⢨âì á ­¥ª®â®àë¬

Yj ,

â®

R(f, g)

an = bm = 1.

®¡à â¨âìáï ¢ 0 ¯® á«¥¤á⢨î 9.27.

ɇǬ

’ ª¨¬

®¡à §®¬, ¯® ¯à¥¤«®¦¥­¨î 9.28

Y (Xi − Yj ),

R(f, g) = A

A ∈ R.

(77)

i,j

® ä®à¬ã« ¬ ‚¨¥â 

Y Y (Xi − Yj ) = g(Xi ). i,j

‘«¥¤®¢ â¥«ì­®, á⥯¥­ì

σj (X1 , . . . , Xn )

¯®

Xi

à §®¬, ¢ (77) á⥯¥­ì ¯¥à¥¬¥­­ëå

Xi , Yj .

Q

i,j (Xi

− Yj )

¯® «î¡®¬ã

à ¢­  1. ®í⮬ã á⥯¥­ì

A

¯®

(78)

i

Xi à ¢­  A ∈ K.

Xi à ¢­  m. ‘ ¤à㣮© áâ®à®­ë, á⥯¥­ì R(f, g) ¯® Xi ­¥ ¬¥­ìè¥ m. ’ ª¨¬ ®¡-

0. €­ «®£¨ç­ë¥ à áá㦤¥­¨ï ¯à¨¬¥­¨¬ë ¤«ï ¢á¥å

®í⮬ã

®«®¦¨¬

X1 = · · · = Xn = 0,

Y1 = · · · = Ym = 1.

’®£¤ 

Y (Xi − Yj ) = (−1)mn ,

¨

σi (0, . . . , 0) = 0,

i,j

  m σj (1, . . . , 1) = . j

Žâá

1 0 0 .............................. 0 0 1 0 .............................. 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0
0 ... 1 0 ........... 0 m . . . . . . . . . . . . . (−1)m 0 ... 0 = (−1)mn . R(f, g)(0, . . . , 0, 1, . . . , 1) = 1 − 1 .. .. .. .. .. .. 0 . . . . . . 0 1 .. .. .. .. .. .. 0 . (−1)m . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (−1)m ®í⮬ã

A = 1.

’¥®à¥¬ 

9.30.

an Disc(f ) = (−1)

„®ª § â¥«ìá⢮.

n(n−1) 2

R(f, f 0 ).

ãáâì

n

f = an X + an−1 X n−1 + · · · + a0 , …᫨

x1 , . . . , xn

{ ª®à­¨

f,

f 0 = nan X n−1 + · · · + a1 .

â® ª ª ¨ ¢ ¯à¥¤ë¤ã饩 ⥮६¥ (á¬. (78))

R(f, g) = an−1 n

n Y

i=1

f 0 (xi ).

3. „ˆ‘ŠˆŒˆ€’ ˆ …‡“‹œ’€’

‚ëç¨á«¨¬

f 0 (xi ).

ˆ¬¥¥¬

f = an

n Y

(X − xi ),

f0 =

i=1 j6=i

i=1

Žâá

f 0 (xi ) = an

n Y n X (X − xj ).

n Y (xi − xj ), j6=i

¨, á«¥¤®¢ â¥«ì­®,

Qn Q R(f, f 0 ) = a2n−1 n i=1 j6=i (xi − xj ) = n(n−1) Q n(n−1) 2 2 a2n−1 (−1) 2 Disc(f ). n j