## Galois groups of finite fields

Each acts in a well-defined way on the finite field , so we obtain a homomorphism

We pause for a moment and derive a few basic properties of , which are in fact general properties of Galois groups for finite fields. Let .

The group contains the element defined by

because and

By Exercise 29 (see Chapter 22), the group is cyclic, so there is an element of order , and . Then if and only if which is the case preciselywhen , so the order of is . Since the order of the automorphism group of a field extension is at most the degree of the extension, we conclude that is generated by . Also, since has order equal to the degree, we conclude that is Galois, with group cyclic of order generated by . (Anther general fact: Up to isomorphism there is exactly one finite field of each degree. Indeed, if there were two of degree , then both could be characterized as the set of roots in the compositum of , hence they would be equal.)

William Stein 2004-05-06