#1

Determine, up to isomorphism, the groups of order 154.

Let \(P\) be a \(p\)-Sylow subgroup of a group \(G\), and \(N\) a normal subgroup of \(G\). Prove that \(P\cap N\) is a \(p\)-Sylow subgroup of \(N\).

Determine the number of \(2\times 2\) nilpotent matrices over a finite field \(\mathbb {F}_q\).

Determine, up to conjugacy, all \(4\times 4\) matrices \(M\) over \(\mathbb {Q}\) with \(M^5=-M^3\) and \(M^3\ne 0\).

Let \(M\) be an \(n\times n\) matrix over \(\mathbb {Q}\) in which each entry is 1. What is the Jordan form of \(M\)?

Let \(f(x)\) be a monic polynomial with integer coefficients, such that \(f(\alpha )=0=f(2\alpha )\) for some complex number \(\alpha \). Prove that \(f(0)\ne 1\).

Let \(R\) be a commutative ring with 1.

1. If a maximal ideal \(\mathfrak {m}\) of \(R\) is principal, prove that there is no ideal \(I\) with \(\mathfrak {m}^2\subsetneq I\subsetneq \mathfrak {m}\).
2. Give an example where \(\mathfrak {m}\) is maximal, but there exists an ideal \(I\) with \(\mathfrak {m}^2\subsetneq I\subsetneq \mathfrak {m}\).

Let \(\alpha \) be a complex root of \(x^6+3\). Set \(K=\mathbb {Q}(\alpha )\).

1. Prove that \(K\) contains a primitive 6-th root of unity.
2. Compute the Galois group of \(K\) over \(\mathbb {Q}\).

Let \(\zeta \) be a primitive 16-th root of unity over a field \(K\). Determine \([K(\zeta ):K]\) when \(K\) is:

1. \(\mathbb {F}_7\)
2. \(\mathbb {F}_9\)
3. \(\mathbb {F}_{17}\)

Explain (preferably in a sentence!) why \(\mathbb {F}_3[x]/(x^2-2)\) and \(\mathbb {F}_3[x]/(x^2-2x-1)\) are isomorphic. Then construct an explicit isomorphism.