1.2 A Simple Problem

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determines a maximal ideal

MA = Ker{/ -+ /(A) : H(U) - C}

and, in fact, every maximal ideal with C as quotient field has this form.

To see this, observe that if M is a maximal ideal with quotient field C,

cj) : H(U) — C is the quotient homomorphism modulo M, and A = j(z),

then z- X belongs to M. But, for each / G H{U), it follows from the power

series expansion of / at A that f{z) — /(A) is divisible by z — A. That is,

^ has a removable singularity at A and so it defines an h G T~i{U)

z — A

such that f(z) - /(A) = (z - X)h(z). Thus, / - /(A) also belongs to M

and cj)(f) - /(A) = /(/ - /(A)) = 0. This means that (j) agrees with the

evaluation homomorphism / — • /(A) and M = M\.

Is every maximal ideal of H(U) of the form M\l No. Let {an} c U

be a sequence of points which has no limit point in U. Let / be the set of

functions in H(U) which vanish at all but finitely many points of {an}. Then

J is a proper ideal of H(U). However, there is no single point at which all

the functions in / vanish. This follows from the fact that, given any discrete

set S of points of £/, there is a function in H(U) which vanishes exactly on

S (a corollary of the Weierstrass theorem, which we will prove later in the

chapter). Thus, no maximal ideal that contains / can be of the form M\.

The following is true, however:

1.2.1 Theorem. Each finitely generated maximal ideal of 7i(U) is of the

form M\ for some A G U.

If a maximal ideal M of H{U) is finitely generated, say by {gi,..., gn},

and if these generators all vanish at some point A G £/, then all the functions

in M vanish at A and we have M C M\. This, of course, implies that

M = MA, since M is a maximal ideal. Thus, Theorem 1.2.1 will be proved

if we can show that: if a finite set of functions {#i,. . . , gn} does not have a

common zero, then it does not generate a proper ideal. That is the content

of the following proposition.

1.2.2 Proposition. If a finite set of functions {gi,. . . , gn} c H(U) has no

common zero in U

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then the equation

(1.2.1) fl9l + '" + fn9n = l

has a solution for / i , . . . , fn G H{U).

The proof of this result will occupy most of the chapter. Along the way,

we shall prove a number of important results from the theory of holomor-

phic functions of one complex variable that are often not covered in a first