University of Utah, Department of Mathematics Algebra 1 Qualifying Exam January 2026

Let \(R := \mathbb {Z}\), and let \(M\) be the cokernel of the map \[ R^4 \xrightarrow { \begin {pmatrix} 2 & 7 & 3 & 4\\ 3 & 8 & 7 & 11\\ 4 & 10 & 10 & 16 \end {pmatrix} } R^3. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Let \(A := \mathbb {R}[x,y]/(x^2+y^2+1)\). Let \(M \subset A\) be a maximal ideal. Determine the field \(A/M\).

Let \(I\) be the ideal \((2,1+\sqrt {-5})\) in the ring \(R := \mathbb {Z}[\sqrt {-5}]\). Prove that \(I\otimes _R I\) is isomorphic to \(2R\).

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.

Let \(M_n(\mathbb {C})\) be the algebra of \(n\) by \(n\) complex matrices. Let \(T \in M_n(\mathbb {C})\) such that \(T^m=1\) for some \(m>0\). Prove that \(T\) is conjugate to a diagonal matrix.