University of Utah, Department of Mathematics May 2018, Algebra Qualifying Exam

Show there is no simple group of order 144.

How many elements of order 3 can a group of order 21 have? List all the possibilities.

Suppose that \(R=\mathbb {C}[x]\), and that \(M\) is the \(R\)-module generated by three elements \(a,b\), and \(c\), modulo the three relations \(a-xc\), \(xa-xb+xc\), and \(xb-(x^2-1)c\). Write \(M\) as a direct sum of cyclic modules.

Determine the possible characteristic and minimal polynomials of an \(n\times n\) matrix over \(\mathbb {C}\) that has rank 1.

Find all the prime ideals of \(\mathbb {Z}[x]/(30,x^3+1)\) and identify which are maximal.

Suppose \(R=\mathbb {F}_2[x,y,z]\). Compute \(\operatorname {Tor}^R_1(R/(x,y),R/(y,z))\).

Determine the irreducible factorization of \(x^5-1\) over the field \(\mathbb {F}_{19}\).

Let \(M/K\) be a Galois extension of degree 700. Prove that there exists an intermediate field \(L\) with \([L:K]=28\). Is \(L/K\) necessarily Galois?

Determine the Galois group of \(x^8-2\) over \(\mathbb {Q}\).

Find a Galois extension of \(\mathbb {Q}\) with Galois group isomorphic to \(\mathbb {Z}/3\times \mathbb {Z}/3\).