6320-2021-08

Find a degree 5 monic polynomial with integer coefficients that is irreducible in \(\mathbb {Q}[x]\) but does not satisfy the Eisenstein criterion for any prime \(p\).

Find a non-abelian finite subgroup of \(\operatorname {GL}_2(\mathbb {C})\) that is not conjugate to a finite subgroup of \(\operatorname {GL}_2(\mathbb {Z})\). Here for a ring \(R\), \(\operatorname {GL}_2(R)\) denotes the group of invertible \(2 \times 2\) matrices with entries in \(R\).

Find a complex number \(\eta \) such that \(\mathbb {Q}(\eta )/\mathbb {Q}\) is Galois with Galois group isomorphic to the dihedral group of order 8.

For \(p\) a prime number, show that \(x^{p^n}-x \in \mathbb {F}_p[x]\) is the product of all irreducible polynomials of degree dividing \(n\) in \(\mathbb {F}_p[x]\).

Is every group of order 480 solvable?