In: Advanced Math
3. Show that the Galois group of (x2 − 3)(x2 + 3) over Q is isomorphic to Z2 × Z2.
4. Let p(x) be an irreducible polynomial of degree n over a finite field K. Show that its Galois group over K is cyclic of order n.
Proof: 3.
There is a factorization
, which implies that the splitting field of
is
.
Since
it follows that
is a group of order
.
There are two automorphisms of
that are the identity on
, namely ,
and
which is determined by
.
And for
,
there are two automorphisms of
that agree with
on
and send
to
.
Therefore, there are automorphisms of
determined by the actions on
by the following table:
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Since there are exactly
elements of
listed, and the order of the group is
, it follows that the listed elements are all of the elements of
the Galois group.
Since
and
it follows that
.
Thus, every nonidentity element of
has order
so the Galois group is isomorphic to
.