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a + b and ab of addition and multiplication (for each of these k = 2).
§2. Fields We start by describing one type of 'sets with operations' as described in § 1 which corresponds most closely to our intuition of numbers. A field is a set K on which two operations are defined, each taking two elements of K into a third; these operations are called addition and multiplication, and the result of applying them to elements a and b is denoted by a + b and ab. The operations are required to satisfy the following conditions: Addition: Commutativity: a + b = b + a; Associativity: a + (b + c) = (a + b) + c; Existence of zero: there exists an element 0 e K such that a + 0 = a for every a (it can be shown that this element is unique); Existence of negative: for any a there exists an element ( — a) such that a + (— a) = 0 (it can be shown that this element is unique). Multiplication: Commutativity: ab = ba; Associativity: a(bc) = (ab)c; Existence of unity: there exists an element 1 e K such that a\ = a for every a (it can be shown that this element is unique); Existence of inverse: for any a / 0 there exists an element a~J such that aa'1 = 1 (it can be shown that for given a, this element is unique). Addition and multiplication: Distributivity: a(b + c) = ab + ac. Finally, we assume that a field does not consist only of the element 0, or equivalently, that 0 # 1.
12
§2. Fields
These conditions taken as a whole, are called the field axioms. The ordinary identities of algebra, such as (a + bf = a2 + lab + b2 or a" 1 -(a+
I)" 1 = a " 1 ( a + l ) ~ 1
follow from the field axioms. We only have to bear in mind that for a natural number n, the expression na means a + a + ••• + a (n times), rather than the product of a with the number n (which may not be in K). Working over an arbitrary field K (that is, assuming that all coordinates, coefficients, and so on appearing in the argument belong to K) provides the most natural context for constructing those parts of linear algebra and analytic geometry not involving lengths, polynomial algebras, rational fractions, and so on. Basic examples of fields are the field of rational numbers, denoted by Q, the field of real numbers U and the field of complex numbers C If the elements of a field K are contained among the elements of a field L and the operations in K and L agree, then we say that K is a subfield of L, and L an extension ofK, and we write K a L. For example, Q c R c C . Example 1. In § 1, in connection with the 'geometry' of Figure 1, we defined operations of addition and multiplication on the set {0,1}. It is easy to check that this is a field, in which i is the zero element and 1 the unity. If we write 0 for (D) and 1 for 11, we see that the multiplication table of Figure 4 is just the rule for multiplying 0 and 1 in a" and b'£>" implies that a' + b' *-* a" + b" and a'b a"fo"; we say that two fields are isomorphic if there exists an isomorphism between them. If L' and L" are isomorphic fields, both of which are extensions of the same field K, and if the isomorphism between them takes each element of K into itself, then we say that it is an isomorphism over K, and that L' and L" are isomorphic over K. An isomorphism of fields K' and K" is denoted by K' s K". If L' and L" are finite fields, then to say that they are isomorphic means that their addition and multiplication tables are the same; that is, they differ only in the notation for the elements of L' and L". The notion of isomorphism for arbitrary fields is similar in meaning. For example, suppose we take some line a and mark a point 0 and a 'unit interval' OE on it; then we can in a geometric way define addition and multiplication on the directed intervals (or vectors) contained in a. Their construction is given in Figures 5-6. In Figure 5, b is an arbitrary line parallel to a and U an arbitrary point on it, OU\\AV and VC\\ UB; then OC = OA + OB. In Figure 6, b is an arbitrary line passing through O, and EU \\ BV and VC || UA; then OC = OA • OB.
Fig. 5
Fig. 6
With this definition of the operations, intervals of the line form a field P; to verify all the axioms is a sequence of nontrivial geometric problems. Taking each interval into a real number, for example an infinite decimal fraction (this is again a process of measurement!), we obtain an isomorphism between P and the real number field U.
14
§2. Fields
Example 3. We return now to the plane curve given by F(x, y) = 0, where F is a polynomial; let C denote the curve itself. Taking C into the set of coefficients of F is one very primitive method of 'coordinatising' C. We now describe another method, which is much more precise and effective. It is not hard to show that any nonconstant polynomial F(x, y) can be factorised as a product of a number of nonconstant polynomials, each of which cannot be factorised any further. If F = Fx • F2... Fk is such a factorisation then our curve with equation F = 0 is the union of k curves with equations Ft = 0, F2 = 0, ..., Fk = 0 respectively. We say that a polynomial which does not factorise as a product of nonconstant polynomials is irreducible. From now on we will assume that F is irreducible. Consider an arbitrary rational function is represented as a ratio of two polynomials:
(p(xy)
2) ——-, Q(x,y) and we suppose that the denominator Q is not divisible by F. Consider ipasa function on points of C only; it is undefined on points (x, y) where both Q(x, y) = 0 and F(x, y) = 0. It can be proved that under the assumptions we have made there are only finitely many such points. In order that our treatment from now on has some content, we assume that the curve C has infinitely many points (that is, we exclude curves such as x2 + y2 = — 1, x4 + y4 = 0 and so on; if we also consider points with complex coordinates, then the assumption is not necessary). Then (p(x, y) defines a function on the set of points of C (for short, we say on C), possibly undefined at a finite number of points—in the same way that the rational function (1) is undefined at the finite number of values of x where the denominator of (1) vanishes. Functions obtained in this way are called rational functions on C. It can be proved that all the rational functions on a curve C form a field (for example, one proves that a function cp defines a nonzero function on C only if P(x, y) is not divisible by F(x, v), and then the function a>~1 =
— satisfies the
P(x,y) condition required for cp, that the denominator is not divisible by F; this proves the existence of the inverse). The field of rational functions on C is denoted by U(C); it is an extension of the real number field U. Considering points with coordinates in an arbitrary field K, it is easy to replace U by K in this construction. Assigning to a curve C the field K(C) is a much more precise method of 'coordinatising' C than the coefficients of its equation. First of all, passing from a coordinate system (x,y) to another system (x',y'\ the equation of a curve changes, but the field K(C) is replaced by an isomorphic field, as one sees easily. Another important point is that an isomorphism of fields K(C) and K(C) establishes important relations between curves C and C. Suppose as a first example that C is the x-axis. Then since the equation of C is y = 0, restricting a function cp to C we must set y = 0 in (2), and we get a rational function of x:
§2. Fields
A and B contains an element x with the property that every element of B can be uniquely written in the form a0 + a1x + ••• + anx"
with at e A
for some n ^ 0. If B' is another such ring, with x' the corresponding element, the correspondence
§3. Commutative Rings
17
annuluses of convergence). With the usual definition of operations on series, these form a field, the field of Laurent series. If we use the same rules to compute the coefficients, we can define the sum and product of two Laurent series, even if these are nowhere convergent. We thus obtain the field of formal Laurent series. One can go further, and consider this construction in the case that the coefficients an belong to an arbitrary field K. The resulting field is called the field of formal Laurent series with coefficients in K, and is denoted by K((z)).
§ 3. Commutative Rings The simplest possible example of 'coordinatisation' is counting, and it leads (once 0 and negative numbers have been introduced) to the integers, which do not form a field. Operations of addition and multiplication are defined on the set of all integers (positive, zero or negative), and these satisfy all the field axioms but one, namely the existence of an inverse element a"1 for every a # 0 (since, for example, j is already not an integer). A set having two operations, called addition and multiplication, satisfying all the field axioms except possibly for the requirement of existence of an inverse element a"1 for every a # 0 is called a commutative ring; it is convenient not to exclude the ring consisting just of the single element 0 from the definition. The field axioms, with the axiom of the existence of an inverse and the condition 0 # 1 omitted will from now on be referred to as the commutative ring axioms. By analogy with fields, we define the notions of a subring A a B of a ring, and isomorphism of two rings A' and A"; in the case that A c A' and A / ^ 5 is reducible, for example 2 + y/^-5 = aft, then N(2 + v ^ 5 ) = N(a)N(fi). But N(2 + ^^5) = 9, and hence there are only three possibilities: (N(a), N(fi)) = (3,3) or (1,9) or (9,1). The first of these is impossible, since 3 cannot be written in the form n2 + 5m2 with n, m integers. In the second P = +1 and in the third a = ± 1, so a or ft is a unit. This proves that 2 + ^/— 5 is irreducible. To say that a ring is not a UFD does not mean to say that it does not have an interesting theory of divisibility. On the contrary, in this case the theory of divisibility becomes especially interesting. We will discuss this in more detail in the following section § 4.
24
§4. Homomorphisms and Ideals
§ 4. Homomorphisms and Ideals A further difference of principle between arbitrary commutative rings and fields is the existence of nontrivial homomorphisms. A homomorphism of a ring A to a ring £ is a map / : A ->• B such that
/ ( a 1 + f l 2 ) = / ( a 1 ) + /(a 2 ), f{a1a2)=f(a1)-f(a2)
and
f(lA)=lB
(we write 1A and lB for the identity elements of A of B). An isomorphism is a homomorphism having an inverse. If a ring has a topology, then usually only continuous homomorphisms are of interest. Typical examples of homomorphisms arise if the rings A and B are realised as rings of functions on sets X and Y (for example, continuous, differentiable or analytic functions, or polynomial functions on an algebraic curve C). A map q>: Y -* X transforms a function F on X into the function q>*F on Y defined by the condition (cp*F)(y) = If cp satisfies the natural conditions for the theory under consideration (that is, if (p is a continuous, differentiable or analytic map, or is given by polynomial expressions) then * is simply the resjriction to Y of functions defined on X. Example 1. If C is a curve, defined by the equation F(x, y) = 0 where F e K[x, y] is an irreducible polynomial, then restriction to C defines a homomorphism K[x,y] -> K[C]. The case which arises most often is when Y is one point of a set X, that is Y = {x0} with x 0 e X; then we are just evaluating a function, taking it into its value at x0. Example 2. If x 0 e C then taking each function of K\_C] into its value at x0 defines a homomorphism K [C] -> K. Example 3. If t? is the ring of continuous functions on [0,1] and x0 e [0,1] then taking a function q> e %> into its value cp(x0) is a homomorphism %? -» R. If A is the ring of functions which are holomorphic in a neighbourhood of 0, then taking
(0) is a homomorphism A-+C. Interpreting the evaluation of a function at a point as a homomorphism has led to a general point of view on the theory of rings, according to which a commutative ring can very often be interpreted as a ring of functions on a set, the 'points' of which correspond to homomorphisms of the original ring into fields. The original example is the ring X[C], where C is an algebraic variety, and from it, the geometric intuition spreads out to more general rings. Thus the concept that 'every geometric object can be coordinatised by some ring of
§ 4. Homomorphisms and Ideals
25
functions on it' is complemented by another, that 'any ring coordinatises some geometric object'. We have already run into these two points of view, the algebraic and functional, in the definition of the polynomial ring in § 3. The relation between the two will gradually become deeper and clearer in what follows. Example 4. Consider the ring A of functions which are holomorphic in the disc \z\ < 1 and continuous for \z\ < 1. In the same way as before, any point z0 with zo\ < 1 defines a homomorphism A -» C, taking a function q> e A into q>(z0). It can be proved that all homomorphisms A-*C over C are provided in this way. Consider the boundary values of functions in A; these are continuous functions on the circle \z\ = 1, whose Fourier coefficients with negative index are all zero, that is, with Fourier expansions of the form £ cne2nil"l>. Since a function f e A is determined by its boundary values, A is isomorphic to the ring of continuous functions on the circle with Fourier series of the indicated type. However, in this interpretation, only the homomorphisms of A corresponding to points of the boundary circle \z\ = 1 are immediately visible. Thus considering the set of all homomorphisms sometimes helps to reestablish the set on which the elements of the ring should naturally be viewed as functions. In the ring of functions which are holomorphic and bounded for \z\ < 1, by no means all homomorphisms are given in terms of points z 0 with |z o | < 1. The study of these is related to delicate questions of the theory of analytic functions. For a Boolean ring (see § 3, Example 9), it is easy to see that the image of a homomorphism cp: A —• F in a field F is a field with two elements. Hence, conversely, any element a e A sends a homomorphism cp to the element q>(a) e F2. This is the idea of the proof of the main theorem on Boolean rings: for M one takes the set of all homomorphisms A -» F2, and A is interpreted as a ring of functions on M with values in F2. Example 5. Let JT be a compact subset of the space C" of n complex variables, and A the ring of functions which are uniform limits of polynomials on Jf. The homomorphisms A -» C over C are not exhausted by those corresponding to points z G Jf; they are in 1-to-l correspondence with points of the socalled polynomial convex hull of Jf, that is with the points z e C such that \f(z)\ < Sup | / | for every polynomial / . mr
Example 6. Suppose we assign to an integer the symbol © if it is even and 1 if it is odd. We get a homomorphism Z -»F 2 of the ring of integers to the field with 2 elements F2 (addition and multiplication tables of which were given in § 1, Figures 3 and 4). Properly speaking, the operations on © and 11 were defined in order that this map should be a homomorphism. Let / : A ->B be a homomorphism of commutative rings. The set of elements f(a) with a e A forms a subring of B, as one sees from the definition of homomorphism; this is called the image of/, and is denoted by I m / or f(A). The set of
26
§ 4. Homomorphisms and Ideals
elements a e A for which f(a) = 0 is called the kernel of/, and denoted by Ker/. If B = Im / then we say that B is a homomorphic image of A. If K e r / = 0 then / is an isomorphism from A to the subring f(A) of B; for if /(a) = /(£>) then it follows from the definition of homomorphism that f(a — b) = 0, that is, a — b e K e r / = 0 and so a = b. Thus / is a 1-to-l correspondence from A to f{A), and hence an isomorphism. This fact draws our attention to the importance of the kernels of homomorphisms. It follows at once from the definitions that if alt a2e K e r / then a1 + a2 e K e r / and if a e Ker/then ax e K e r / for any x e A. We say that a nonempty subset / of a ring A is an ideal if it satisfies these two properties, that is,
a1, a2 e / =>ax + a2 e /, and ael=>axel
for any x e A.
Thus the kernel of any homomorphism is an ideal. A universal method of constructing ideals is as follows. For an arbitrary set {ax} of elements of A, consider the set / of elements which can be represented in the form Yjx*ax f° r some x^e A (we assume that only a finite number of nonzero terms appears in each sum). Then / is an ideal; it is called the ideal generated by {aA}. Most commonly the set {ax} is finite. An ideal / = (a) generated by a single element is called a principal ideal. If a divides b then (b) c (a). A field K has only two ideals, (0) and (1) = K. For if / c K is an ideal of K and 0 # a e I then / a aa~lb = b for any b e K, and hence I = K (this is another way of saying that the theory of divisibility is trivial in a field). It follows from this that any homomorphism K - » B from a field is an isomorphism with some subfieldofB. Conversely, if a commutative ring A does not have any ideals other than (0) and (1), and 0 ^ 1 then A is a field. Indeed, then for any element a / 0 w e must have (a) = A, and in particular 1 e (a), so that 1 = ab for some b e A, and a has an inverse. In the ring of integers Z, any ideal I is principal: it is easy to see that if / # (0) then / = (n), where n is the smallest positive integer contained in /. The same is true of the ring X[x]; here any ideal / is of the form / = (/(x)), where f{x) is a polynomial of smallest degree contained in /. In the ring K[x,y\ it is easy to see that the ideal / of polynomials without constant term is not principal; it is of the form (x,y). An integral domain in which every ideal is principal is called a principal ideal domain (PID). It is not by chance that the rings Z and K [x] are unique factorisation domains: one can prove that any PID is a UFD. But the example of K [x, y] shows that there are more UFDs than PIDs. In exactly the same way, the ring &„ of functions of n > 1 complex variables which are holomorphic at the origin (§ 3, Example 7) is a UFD but not a PID. The study of ideals in this ring plays an important role in the study of local analytic varieties, defined in a neighbourhood of the origin by equations fx = 0,..., fm = 0 (with ft e &n). The representation of such varieties as a union of irreducibles, the notion of their dimension, and so on, are based on properties of these ideals.
§4. Homomorphisms and Ideals
27
Example 7. In the ring ^ of continuous functions on the interval, taking a function
(x) as x —• x0 (for example y/\(p(x)\ is not contained in the ideal (m e IXo of functions in it. Another example of a similar nature can be obtained in the ring & of germs of C00 functions at 0 on the line (by definition two functions defined the same germ at 0 if they are equal in some neighbourhood of 0). The ideal Mn of germs of functions which vanish at 0 together with all of their derivatives of order ^ n is principal, equal to (x" +1 ), but the ideal Mx of germs of functions all of whose derivatives vanish at 0 (such as e~1/x2) is not generated by any finite system of functions, as can be proved. In any case, the extent to which these examples carry conviction should not be exaggerated: it is more natural to use the topology of the ring ^ of continuous functions, and consider ideals topologically generated by functions q>u ..., q>m, that is, the closure of the ideal (q>l,...,q>m). In this topological sense, any ideal of 0
then
nx = x + ••• + x
(n times)
and if n = —m with m > 0 then nx = — (mx). Thus M is just an Abelian group 2 written additively. We omit the definitions of isomorphism and submodule, which repeat word for word the definition of isomorphism and subspace for vector spaces. An isomorphism of modules M and N is written M s N. Example 7. Any differential r-form on n-dimensional Euclidean space R" can be uniquely written in the form aH...irdxh
Z ••i
A
"••
A
dx
w-
< • • • < > , .
where ati_mir belongs to the ring A of functions on R" (differentiable, real analytic or complex analytic, see Example 2). Hence the module of differential forms is isomorphic to A^r), where I
I is the binomial coefficient.
Example 8. Consider the polynomial ring C[x 1,...,xn] as a module M over itself (Example 1); on the other hand, consider it as a module over the ring of differential operators with constant coefficients (Example 4). Since this ring is isomorphic to the polynomial ring, we get a new module N over C [ x t , . . . , xn~]. These modules are not isomorphic; in fact for any m' e N there exists a non-zero element ae€,[xl,...,xn] such that am' = 0 (take a to be any differential operator of sufficiently high order). But since C[Xi,..., x n ] is an integral domain, it follows that in M,am = 0 implies that a = 0 or m = 0. In a series of cases, Fourier transform establishes an isomorphism of modules M and N over the ring A = C [ x x , . . . , x n ], where M and N are modules consisting of functions, and A acts on M by multiplication, and on N via the isomorphism
For example, this is the case if M = N is the space of Cx functions F(xt,..., for which
xn)
2 We assume that the reader knows the definition of a group and of an Abelian group; these will be repeated in §12.
36
§5. Modules
i
1
Y
•'
n
" dx(>...dx£"
is bounded for all a 5* 0, ft > 0. Recalling the definition of § 4, we can now say that an ideal of a ring A is a submodule of A, \fA is considered as a module over itself (as in Example 1). Ideals which are distinct as subsets of A can be isomorphic as A -modules. For example, an ideal / of an integral domain A is isomorphic to A as an A-module if and only if it is principal (because if / = (/) then a \—* ai is the required homomorphism; conversely, if (a\) = aq>{\) = ai, that is / = () = 0, (that is (/(
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