1.7. Generators of extension fields.

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Let E be an extension field of F, and let S be a

subset of E. The intersection of all the subrings of E containing F and S is again a subring

of E (containing F and S). We call it the subring of E generated by F and S, and we write

it F[S].

6 J.S. MILNE

Lemma 1.13. The ring F[S] consists of all the elements of E that can be written as finite

sums of the form

_

ai1···inαi1

1

・ ・ ・ αin

n, ai1···in

∈ F, αi ∈ S. (*)

Proof. Let R be the set of all such elements;it is easy to check that R is a ring containing

F and S, and that any ring containing F and S contains R;t herefore R equals F[S].

Note that the expression of an element in the form (*) will not be unique in general. When

S = {α1, ..., αn}, we write F[α1, ..., αn] for F[S].

Lemma 1.14. Let E ⊃ R ⊃ F with E and F fields and R a ring. If R is finite-dimensional

when regarded as an F-vector space, then it is a field.

Proof. Let α be a nonzero element of R—we have to show that α is invertible. The map

x → αx : R → R is an injective F-linear map, and is therefore surjective. In particular,

there is an element β ∈ R such that αβ = 1.

Example 1.15. An element of Q[π], π = 3.14159..., can be written uniquely as a finite

sum

a0 + a1π + a2π2 + ・ ・ ・, ai ∈ Q.

An element of Q[i] can be written uniquely in the form a + bi, a, b ∈ Q. (Everything

considered in C.)

Let E again be an extension field of F and S a subset of E. The subfield F(S) of E

generated by F and S is the intersection of all subfields of E containing F and S. It is

equal to the field of fractions of F[S] (since this is a field containing F and S, and is the

smallest such field). Lemma 1.14 shows that F[S] is sometimes already a field, in which case

F(S) = F[S]. We write F(α1, ..., αn) for F(S) when S = {α1, ..., αn}.

Thus: F[α1, . . . , αn] consists of all elements of E that can be expressed as polynomials in

the αi with coefficients in F, and F(α1, . . . , αn) consists of all elements of E that can be

expressed as quotients of two such polynomials.

Example 1.16. An element of Q(π) can be expressed as a quotient

g(π)/h(π), g(X), h(X) ∈ Q[X], h(π) _= 0.

The ring Q[i] is already a field.

An extension E of F is said to be simple if E = F(α) some α ∈ E. For example, Q(π)

and Q[i] are simple extensions of Q.

When F and F_ are subfields of E, then we write F ・ F_ for F(F_)(= F_(F)), and we call

it the composite of F and F_. It is the smallest subfield of E containing both F and F_.

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