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Consider the symmetric group \(S_p\) for \(p\) an odd prime. Let \(C\) be the centralizer of the \(p\)-cycle \(\sigma =(1\ 2\ \ldots \ p)\) and let \(N\) be the normalizer of \(\langle \sigma \rangle \) in \(S_p\). What is the order of each of the groups \(C\) and \(N\)? Are they abelian?

Let \(U\) be the subgroup of \(\operatorname {GL}_2(\mathbb {Z}/p\mathbb {Z})\) consisting of upper triangular matrices. Is \(U\) solvable? Prove or disprove.

A group of order 81 acts on a set \(X\) with 30 elements. Show that some element of \(X\) is fixed by at least 27 elements. How many elements of \(X\) are fixed by precisely 3 elements, assuming there is at least one such element?

Let \(S\) be a finite subset of \(\mathbb {C}\). Let \(X\) be the set of \(n\times n\) complex matrices all of whose eigenvalues lie in \(S\). Consider the conjugation action of \(\operatorname {GL}_n(\mathbb {C})\) acting on \(X\). Prove that this action has finitely many orbits. If \(n=3\), find a formula for the number of orbits in terms of the cardinality of \(S\).

Let \(R\) be a commutative ring. If for each prime ideal \(\mathfrak {p}\) of \(R\), the local ring \(R_{\mathfrak {p}}\) contains no nonzero nilpotent elements, prove that \(R\) contains no nonzero nilpotent elements.

Let \(\mathfrak {a}\) be the ideal of the polynomial ring \(\mathbb {R}[x]\) that is generated by \(x^3-5x^2+6x-2\) and \(x^4-4x^3+3x^2-4x+2\). Is \(\mathfrak {a}\) a principal ideal? Is it prime? Is it maximal?

Let \(\mathbb {F}\) be a field with 81 elements. What is the number of roots of \(x^{25}-1=0\) in \(\mathbb {F}\)?

Suppose \(f(x)\in \mathbb {Q}[x]\) is an irreducible polynomial with complex roots \(\alpha _1,\ldots ,\alpha _n\). Prove that \(\alpha _i-\alpha _j\notin \mathbb {Q}\) for all \(i\ne j\).

Let \(K\) be a subfield of a field \(L\). If \(\sigma :L\longrightarrow L\) is a homomorphism that fixes all elements of \(K\), is \(\sigma \) necessarily an automorphism of \(L\)? Justify your answer.

Let \(L=\mathbb {Q}(e^{2\pi i/8})\). Determine all subgroups of the Galois group \(\operatorname {Gal}(L/\mathbb {Q})\), and the corresponding intermediate fields.