6320-2023-08

Prove that the alternating group \(A_5\) is simple.

Find all the groups \(G\) of order 18.

How many irreducible polynomials of degree 12 in \(\mathbb {F}_p[x]\) are there?

Prove that the Galois group of an irreducible polynomial \(f(x) \in \mathbb {Q}[x]\) of degree \(p\) (a prime) with one pair of complex conjugate roots and \(p-2\) real roots is the full symmetric group \(S_p\).

Find the Galois group of \(x^8-2 \in \mathbb {Q}[x]\). In particular, prove that it has order 16 and is not the dihedral group.