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

Prove that \(a^{561}\equiv a\pmod {561}\) for each integer \(a\).

Determine the integers \(n\) for which there exists a surjective homomorphism of symmetric groups \(S_{n+1}\longrightarrow S_n\).

Suppose \(G\) is a finite group with \(\operatorname {Aut}G\) solvable. Prove that \(G\) is solvable.

Suppose a finite group \(G\ne \{e\}\) has \(c\) conjugacy classes. Prove that \(G\) contains an element other than the identity of order at most \(c\).

Let \(R=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^2\longrightarrow R^3 \qquad \text {given by}\qquad \begin {pmatrix} x+1 & 0\\ 2x & x^2\\ 1 & 1 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Determine, up to conjugacy, all real \(3\times 3\) matrices \(A\) satisfying \(A^8=I\) and \(A^4\ne I\).

Let \(p\) be a prime number and \(n\) a positive integer. How many elements \(\alpha \) are in \(\mathbb {F}_{p^n}\) such that \(\mathbb {F}_p(\alpha )=\mathbb {F}_{p^6}\)?

Let \(k\) be a field and \(k(x)\) the field of rational functions over \(k\), in the indeterminate \(x\). Consider the automorphisms \(\sigma \) and \(\tau \) of \(k(x)/k\) defined by \[ \sigma (x)=\frac {1}{1-x}\qquad \text {and}\qquad \tau (x)=\frac {1}{x}. \]

1. Prove that the subgroup \(G=\langle \sigma ,\tau \rangle \) of \(\operatorname {Aut}(k(x)/k)\) is isomorphic to \(S_3\).
2. Prove that the fixed subfield under the action of \(G\) equals \(k(t)\), where \[ t=\frac {(x^2-x+1)^3}{x^2(x-1)^2}. \]

Let \(n\) a positive integer, and let \(\mathbf {x}:=x_1,\ldots ,x_n\) be indeterminates over \(\mathbb {Q}\). For each positive integer \(i\) set \[ \alpha _i=x_1^i+\cdots +x_n^i. \]

1. Clearly \(\mathbb {Q}(\alpha _1,\ldots ,\alpha _n)\subseteq \mathbb {Q}(s_1,\ldots ,s_n)\), the field of symmetric functions in the \(\mathbf {x}\). What is the degree of this extension?
2. Prove the characteristic polynomial of an \(n\times n\) matrix \(A\) over \(\mathbb {Q}\) is completely determined by the elements \(\operatorname {trace}(A),\ldots ,\operatorname {trace}(A^n)\).

Let \(E:=\mathbb {Q}(\zeta +\zeta ^{-1})\) where \(\zeta \in \mathbb {C}\) is a primitive \(n\)th root of 1, for some integer \(n\ge 3\). Determine the Galois group of \(E\) over \(\mathbb {Q}\).