University of Utah, Department of Mathematics January 2019, Algebra Qualifying Exam

Determine, up to isomorphism, the groups of order 20.

Is every automorphism of the alternating group \(A_4\) an inner automorphism?

Let \(S_n\) denote the symmetric group on \(n\) elements. Does there exist an injective homomorphism \(S_6\longrightarrow S_5\times S_5\)?

Let \(n\ge 2\) be an integer. Prove that the natural map \(SL_2(\mathbb {Z})\longrightarrow SL_2(\mathbb {Z}/n)\) is surjective.

Determine, up to conjugacy, all \(3\times 3\) matrices \(M\) over \(\mathbb {Q}\) that satisfy \(M^3=2M^2\).

Let \(M\) and \(N\) be \(n\times n\) matrices over \(\mathbb {C}\) with \(MN-NM=M\). Prove that \(M^kN-NM^k=kM^k\) for each \(k\ge 1\). Prove that \(M\) is nilpotent.

Let \(R=\mathbb {Q}[x]\) and consider the submodule \(M\) of \(R^2\) generated by the elements \((x^2-1,x-1)\) and \((x^2+x,x)\). Write \(M\) as a direct sum of cyclic modules.

Identify all the prime ideals in the ring \((\mathbb {Z}/14)[x,y]/(2y-1,x^2+x-y^2)\).

Suppose that \(R\) is a commutative ring with unity, and \(N\subseteq M\) are \(R\)-modules. If \(\operatorname {Ext}^1(M/N,N)=0\), prove that the inclusion \(N\hookrightarrow M\) is split.

Let \(K\) be a field, and let \(L\) be the splitting field of an irreducible and separable polynomial \(g(x)\in K[x]\). If \(\operatorname {Gal}(L/K)\) is abelian, and \(\alpha \in L\) is a root of \(g(x)\), prove that \(L=K(\alpha )\).