Question

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.

Answer 1

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:

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 .