Question

In: Advanced Math

3. Show that the Galois group of (x2 − 3)(x2 + 3) over Q is isomorphic...

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.

Expert Solution

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 .

Colby Messinger answered 1 year ago

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