Basic Algebra - Computer Graphics and Geometric Modeling - page 850

If A consists of a single element a, then we shall usually write k(a) instead of k(A).

It is easy to show that the intersection of fields is a field and so k(A) is a well-
defined subfield.

Definition. If K is an extension of a field k and if K = k(a) for some element a, then
K is called a simple extension of k and the element a is called a primitive element of
K over k.

Definition. If k is a field, then k(X) will denote the field of quotients , or quotient
field , of k[X]. More generally, k(X 1 ,X 2 ,...,X n ) will denote the field of quotients of
k[X 1 ,X 2 ,...,X n ].

There are natural inclusions

( à ( ) à ◊◊◊à ( )

kkX Ã kXX , kXX , ,..., X n .

1 1 2 1 2

B.8.14. Theorem. Let k be a subfield of a field K. If a ΠK, then either

() ª () ka kX

or a is a root of an irreducible polynomial f(X) in k[X] and

() ª

[] < ( >.

ka kX f X

Proof. See [Mill58]. Note that <f(X)> is a maximal ideal and so k[X]/<f(X)> is a field
by Theorem B.8.5.

B.8.15. Theorem. Let K and K¢ be subfields of fields E and E¢, respectively, and let
j :K Æ K¢ be an isomorphism. Let

() =

n

Œ [] pX a + a X + ◊◊◊+ a X KX

0 1 n

and

n

¢ () = () + () + ◊◊◊+ () Œ []

pX j a j a X j a X KX .

0 1 n

Assume that p(X) is irreducible. If c Œ E and c¢ŒE¢ are roots of the polynomials p(X)
and p¢(X), respectively, then the isomorphism j extends to a unique isomorphism
j¢ : K(c) Æ K¢(c¢) such that j¢(c) = c¢.

Proof. See [Dean66].

B.9 The Complex Numbers

This section will simply define the field of complex numbers. Complex numbers play
an essential role in many areas of mathematics. Appendix E will discuss some of their
nontrivial properties, especially as they relate to analysis.