# A. Extension Fields.

If E is a field and F a subset of E which, under the operations of addition and multiplication in E, itself forms a field, that is, if F is a subfield of E, then we shall call E an extension of F. The relation of being an extension of F will be briefly designated by F C E. If a, fi, y, . . . are elements of E, then by F(a, /3, y, . . . ) we shall mean the set of elements in E which can be expressed as quotients of polynomials in a, y, . . with coefficients in F. It is clear that is a field and is the smallest extension of F which contains the elements or the obtained after the adjunction of the elements

field generated out of F by the elements a, y, . . . . In the sequel all fields will be assumed commutative.

If F C E, then ignoring the operation of multiplication defined between the elements of E, we may consider E as a vector space over F. By the degree of E over F, written (E/F), we shall mean the dimension of the vector space E over F. If (E/F) is finite, E will be called a finite extension.

THEOREM 6. If F, B, E are three fields such that F C B C E, then be elements or L which are linearly be elements independent with respect to B and let of B which are independent with respect to F. Then the products C^ Aj where

are elements of E which are then independent with respect to F. For if is a linear combination of the A, with coefficients in B and because the A. were independent with respect to B we have then requires that each

The independence of the Ct with respect to F

Since there are r . s elements CiAj. we

have shown that for

the degree ( E/F )

If one of the latter numbers

>r . s. Therefore,

is infinite, the theorem follows. If both (E/B) and (B/F) are finite, say r and s respectively, we may suppose that the Aj and the Cj are generating systems of E and B respectively, and we show that the set of products C. Aj is a generating system of E over F. Each A (E can be expressed linearly in terms of the A ^ with coefficients in B. Thus,

Moreover,

being an element of B can be expressed linearly with coefficients in F in terms of the C., i.e., form an independent generating system of E over F.