University of Utah, Department of Mathematics January 2017, Algebra Qualifying Exam

Prove that there is no simple group of order 132.

How many elements of the symmetric group \(S_n\) commute with a fixed \(k\)-cycle in \(S_n\)?

Suppose \(A\) is a finitely generated abelian group such that \(A\otimes _{\mathbb {Z}}\mathbb {F}=0\) for all finite fields \(\mathbb {F}\). Show that \(A=0\).

Let \(G\) be a finite \(p\)-group, for \(p\) a prime integer. Let \(e\) denote the identity element, and \(Z\) the center of \(G\). If \(N\ne \{e\}\) is normal, prove that \(N\cap Z\ne \{e\}\).

Let \(A\) be an \(m\times n\) real matrix of rank 1. Prove that \(A=BC\), for \(B\) a column matrix, and \(C\) a row matrix.

Describe, up to similarity, all \(8\times 8\) matrices with minimal polynomial \(x^2(x-1)^3\).

Let \(\mathbb {F}_5\) be the field with 5 elements. Construct an explicit isomorphism between the fields \(\mathbb {F}_5[x]/(x^2-2)\) and \(\mathbb {F}_5[x]/(x^2-3)\).

Let \(K=\mathbb {Q}(\pi )\), where \(\pi \) is the ratio of the circumference of a circle to its diameter. You may assume without proof that \(\pi \) is transcendental over \(\mathbb {Q}\).

1. Show that \(f(x)=x^3-\pi \) is irreducible over \(K\).
2. Let \(L\) denote the splitting field of \(f(x)\) over \(K\). Compute \([L:K]\) and \(\operatorname {Gal}(L/K)\).

Let \(a\) be an integer and \(p\) a prime number such that \(x^p-a\) is reducible in \(\mathbb {Q}[x]\). Prove that \(x^p-a\) has root in \(\mathbb {Q}\).

Let \(a,b,c,d\) integers such that \[ \det (A)\ne 0\qquad \text {where}\qquad A=\begin {pmatrix} a & b\\ c & d \end {pmatrix}. \] Let \(x\) and \(y\) be indeterminates, and set \(L=\mathbb {C}(x,y)\) and \(K=\mathbb {C}(x^a y^b,x^c y^d)\).

1. Prove that \(L\) is a finite extension of \(K\) and that \((L:K)=|\det (A)|\).
2. Show that \(L/K\) is Galois, and compute the Galois group.