University of Utah, Department of Mathematics January 2020, Algebra Qualifying Exam

Prove that there are at most 5 groups of order \(2\cdot 7^2\) up to isomorphism.

Suppose \((G,+)\) is a finite Abelian group of order \(3^3\cdot 2^3\). Suppose that \(6\cdot G=\{6x\mid x\in G\}\) has 6 elements. Classify \(G\) as a direct sum of cyclic groups.

Prove that there are no simple groups of order \(2^7\cdot 3^2\).

Let \(R:=\mathbb {Z}[x]\) and \(I\) the ideal \((3,x)\). How many elements are in \(\operatorname {Ext}^i(R/I,R)\) for \(i=0,1,2\)?

Consider \(R:=\mathbb {Q}[x]\) and the matrix: \[ A= \begin {bmatrix} x-1 & 0\\ 1-x & x^2 \end {bmatrix}. \] Let \(M\) be the cokernel of the map \(R^2\longrightarrow R^2\) given by multiplication by \(A\). Compute the rank of the \(\mathbb {Q}\)-vector space \(\operatorname {Hom}_R(M,R/(x^2))\).

Let \(n\) be a positive integer, \(V\) a \(\mathbb {C}\)-vector space of rank \(n\), and \(T:V\longrightarrow V\) a \(\mathbb {C}\)-linear map with the property that for each \(c\in \mathbb {C}\) the eigen space \(\{v\in V\mid T(v)=cv\}\) has rank at most 1. Prove that there exists a \(w\in V\) such that the set \(\{w,T(w),\ldots ,T^{n-1}(w)\}\) is a basis for \(V\).

With \(\alpha =\sqrt {6+\sqrt {11}}\), prove that the extension \(\mathbb {Q}[\alpha ]/\mathbb {Q}\) is Galois and compute its Galois group.

Describe a primitive generator for the degree two extension of \(\mathbb {Q}\) inside the extension \(\mathbb {Q}(\zeta )\), where \(\zeta :=e^{2\pi i/7}\). Justify that the extension you identify is of degree two.

Describe all the prime ideals in the ring \(\mathbb {Z}[x]/(x^3+1,6)\).

For each positive integer \(n\) let \(R_n=\mathbb {Z}[2^{1/2},\ldots ,n^{1/n}]\) viewed as a subring of the field of real numbers. Prove that the ring \(R:=\bigcup _{n\ge 1}R_n\) is not Noetherian.