##

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 general properties of Galois groups for finite fields.
Let
.
The group
contains the element defined by

because
and

The group
is cyclic (see proof of Lemma 8.1.7),
so there is an element
of order
, and
. Then
if
and only if
which is the case precisely when
, 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 . (Another 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
2012-09-24