University of Utah, Department of Mathematics January 2015, Algebra Qualifying Exam

Let \(\sigma \) be a 5-cycle in \(S_5\). How many elements of \(S_5\) commute with \(\sigma \)? How many elements of the subgroup \(A_5\) commute with \(\sigma \)? How many conjugacy classes of 5 cycles are there in \(A_5\)?

Prove that, up to isomorphism, there are at most four groups of order 30.

Let \(H\) be a proper subgroup of a finite group \(G\). Prove that \(\displaystyle \bigcup _{g\in G}gHg^{-1}\) does not equal \(G\).

Set \(G=GL_2(\mathbb {C})\). Construct a proper subgroup \(H\) of \(G\) such that \(\displaystyle \bigcup _{g\in G}gHg^{-1}\) equals \(G\).

Determine, up to conjugacy, the elements of order 4 in \(GL_3(\mathbb {Q})\).

Let \(M\) be \(3\times 3\) matrix over \(\mathbb {C}\) with \(\operatorname {trace}(M^k)=0\) for \(k=1,2,3\). Prove that \(M\) is nilpotent.

Consider the polynomial ring \(\mathbb {Q}[x,y]\) where \(x,y\) are indeterminates over \(\mathbb {Q}\). Determine a finite generating set for the ideal of polynomials \(f(x,y)\) with \(f(i,i)=0\).

Suppose \(K\subset L\subset M\) are fields with \([M:L]=2=[L:K]\). Prove that \(M=K(\alpha )\), where \(\alpha \) is a root of an irreducible polynomial in \(K[x]\) of the form \(x^4+bx^2+c\).

Prove that \(\mathbb {Q}(\sqrt {5+\sqrt {5}})\) is Galois over \(\mathbb {Q}\), and compute the Galois group.

Let \(K\) be a field of characteristic \(p>0\), and let \(t\) be an indeterminate. Consider the automorphism \(\sigma \in \operatorname {Aut}_K K(t)\) with \(\sigma (t)=t+1\). Determine the subfield of \(K(t)\) that is fixed by \(\sigma \).