Some theorems concerning properly irregular cyclomatic fields

MATHEMATICS: H. S. VANDIVER

202

PROC. N. A. S.

It is seen that H(s) (, f) is symbolically the expansion of (

+

bql bpi

@-

bq2 6p2

which suggests a method of extension to functions of three or more pairs of conjugate variables. Proofs and further applications of the formulas appearing in this paper will be published elsewhere. 1 Born and Jordan, Zeits. Physik, 34, 1925 (858-888); Born, Heisenberg and Jordan, Ibid., 35, 1926 (557-615); Born, M., "Problems of Atomic Dynamics," M. I. T. Lectures, 1926. 2 Born, loc. cit., p. 79. 3 Born and Jordan, loc. cit., p. 873. 4 Dirac, P. A. M., Proc. Roy. Soc., 110, 1926, p. 566. c Born, loc. cit., p. 94.

SOME THEOREMS CONCERNING PROPERLY IRREGULAR CYCLOTOMIC FIELDS By H. S. VANDVBR DEPARTMENT OF PURU MATrnrMATICS, UNIVURSITY OF TuXAS Communicated January 23, 1929

Let

2iw

¢=el

and

r) (1_rr)

1

=

(1- )(li(1 -v)(1 --)

C.

where r is a primitive root of the prime I and set E

= r

+sr

-2n

+s2r

-4n

+...+s

I- 3 2 r

n(l

-

3),

the symbol s representing the substitution (v/It) in the notation of the Kronecker-Hilbert symbolic powers; n = 1,2,... (I - 3)/2. If the field k(r) is irregular and none of the above units is the l-th power of a unit in k(r), then this field is said to be properly irregular. I shall prove in another paper' that if k(¢) is properly irregular then the second factor of the class number of this field is prime to 1.2 If co is an integer in k(r) and q is an ideal prime then -

co N(q)

I

(mod q) where N(q) is the norm of q in k(t) and (X) is prime to q. Also there is one and but one integer b in the set 0, 1, . . ., I - 1, such that

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MA THEMA TICS: H. S. VANDIVER N(q)- 1

203

rb (mod q)

Iz

provided q is prime to (1). We write

In the present note I shall indicate how these power characters may be determined, if k(r) is a properly irregular cyclotomic field and co is a singular number in k(r), and q is any ideal in k(r). An integer X in the field k(r) is said to be singular if and only if (X) is the l-th power of some ideal in the field not principal. First let En = a), which is included among the singular numbers. Now, if a is a rational integer, 0 < a < -1, r~it

ind E,,

ind En 32(1

+ al - 2n

where where

-

_

1

(a

+

~~{ En}

1a

(X) = E h

1)1 =

do72 log 4i/'(ed')(1 mod 1; (1)

rind

X(a +1)n + ind(g& + 1)

q is an ideal prime in k(r) whose norm is q', g is a primitive root of q, h ranges over the integers 0, 1, 2, . . ., -q 2, excepting (q' - 1)/2 if q is odd and excepting zero if q is even. The primitive root g is selected so that = (mod q). g(-L1)/l 9 If g, (mod q)k A + 1 0 < k < q' - 1, then we write

ind (gh + 1) = k.

The symbol

do -2n log10Ia(e') dvt - 2n means that the (I- 2n)th derivative of log *a(ev) is taken with respect to v and v = 0 substituted in the result, e being the Napierian base. Let 0(ev) and 41(ev) be two rational functions of e' with rational integral coefficients connected by the relation

(e)= +(e) +

X

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PRoc. N. A. S.

such that

441) =+(1() (mod +1) and

4(1)

then

+(a)

0

(mod 1),

=

)1(a),

and we may prove that

dos

do log O(ev)

log 01(ev) dvml

(mod £+)

(2)

We also have

06ar=

(2a)

X q'c

where c ranges over the integers in the set 1,2,... I-1, such that cc' = 1(mod 1),

where

I - ac + I -c' I > l. is the least positive residue of x, modulo I and c' is in the set, 1, 2, I - 1; also, the qc represents the ideal obtained from q by the substitution (r/¢). Assume that the q belongs to the exponent jei - 1 where j is prime to I so that q'1$ 1('-1) = (X), where w is an integer in kQ(), such that X -- 1 (mod (1 - a)). Raising both sides of (2a) to the power jl' - '(1 - 1) we obtain x

(*ar)i

=1 X CcoV1

(

where I is a unit in k(t). It is easily proved that v is a power of r so that we have the identity

(*a(eTl

(I

-

1) =

ek' + X

(e')

X V

where V is an expression of the type Z

b~evs,

where s ranges over a finite number of positive or negative integers or zero and the bs are rational integers. By (2) (I-

1)jl' -1 do log IW(e) dvm

w¢(e') +mo;1)

olog ed_(o

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205

and (l_l~js -1

Edo' log cs(,?) C

dv""

d"log, w(e') (o

)

dvc After a number of transformations the expression E c"t reduces to a C form which yields from (1)

ind.

- B1~

En

2j'.

- 2Pog (mod 1) do'w(0e) d" xl

(3)

-

where t = (2n - 1)0i Since the En, n = 1,2,3, ..., (I - 3)/2, form a system of independent units in k(r), then any unit in this field may-be expressed as products of powers of these E's, the exponents being fractions whose denominators are prime to l since k(r) is a properly irregular cyclotomic field. Hence the power character of any unit in the field k(r) with respect to any prime ideal q prime to I may be expressed by means of (3). Suppose now that 0 is a singular integer in k(r) which is not necessarily a unit. Since the field is properly irregular, the second factor of the class number is prime to I so that if q is a prime ideal in k(r) prime to 0 and I then (qq- 09 = () where t is an integer in k(r); g is a rational integer prime to 1, and q. 1 represents the ideal obtained from q by the substitution (r/- 1), Hence by the law of reciprocity we have {j} {t}{g}

{qq l}

I = (1 I).

To reduce the symbol {qq0 1} we note that this may be written as

(!@) But since

(e )-1

e-1 =VV

(4)

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206

where v is a unit in k(r) and X an integer in that field, we have I

#@1=X1

so that (4) gives

and the last symbol on the right-hand side is determined by means of (3); hence {O/q) is completely determined. The relation (3) enables us to determine the composition of the irregular class group in a properly irregular cyclotomic field, that is, we may determine directly by the use of rational integers only, the invariants of this group as well as fundamental properties of the basis. We may prove the TIEmoi M.-Consider the Bernoulli numbers *B1, B2, . . ., B I and let 2

the distinct Bernoulli numbers in this group whose numerators are divisible by I be

Bass1 Base*

Bac

Further, let the Bernoulli number lhi(2ai 1) + -

Bd;d =

1

2

be divisible by lth but not by lhi + 1. Then it is possible to find a basis for the irregular class group in the properly irregular cyclotomic field k(r)

Cl, C2... C clhi

Blhi#-1

and such that

P

- 2a

i = 1, 2, ..., e.

Corresponding to this basis there exists a basis for the singular primary numbers in k(r). It consists of the units 1-3

rI -1- 2n I+ST rI- I- 4n 2r 1 1G 3)m -

1

En

=

e

n

=

a,, a2,

+

+...+s

.. .,

a,.

If Is is an ideal belonging to the class Ci and hi= ()

T

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with co = bo + bi r + b2r2 + b

-

2 -2

and

c(ex) = bo + b e' ± . . . + bi-2ev(l- 2) Then, ifk < - 1, [

log oi(e L dVk J

-0 (mod

1)

for k $ £ - 2ai, and

L

- lo 2ai do

= ; 0 (mod 1) i = 1, 2,. .., e.

We may deduce from this theorem the following CORROLLARY. If (X) is the ith power of an ideal in the properly irregular field k(r) then co is a singular number if and only if

[dl - 2a log oI(e)

g

0 (mod 1)

i = 11,2y... ,e,

for at least one value of i. Many known results concerning cyclotomic fields are included in the above results. For example, certain theorems of Takagi.4 Each of the members of the bases of singular primary numbers in the field, the specific units given above, defines an absolute class field of k(r). Through some computations carried out recently, Miss E. T. Stafford and the writer have determined a number of properly irregular cyclotomic fields, namely, for I = 37, 59, 67, 101, 103, 131, 149 and 157. The work was carried through for all primes I less than 200. No irregular fields other than properly irregular fields were discovered. Full details are given in several papers to be presented for publication in the Transactions of the American Mathematical Society. 1 Bull. Amer. Math. Soc., 1929. Synthetische Zdhlentheorie, 2nd edition, p. 223; Bull. Nat. Res. Coun., No. 62, February, 1928, Report of the Committee on Algebraic Numbers, II, pp. 34, 44 45. 3 Kummer, J. Math., 44, 1852, 121-130; Vandiver, Ann. Math., 1929. 4 Takagi, J. reine angew. Math., 157, 236 (1927). 2 Fueter,