6310-2021-01

Let \(d\) be the greatest common divisor of positive integers \(n\) and \(m\). Prove that

Let \(M\) be the cokernel of the map \[ \mathbb {Z}^3\xrightarrow {\begin {pmatrix}2&2&4\\2&4&6\\2&6&8\end {pmatrix}}\mathbb {Z}^3. \]

Write \(M\) as a direct sum of cyclic \(\mathbb {Z}\)-modules.

Let \(I_k\) denote the \(k\times k\) identity matrix. Suppose \(M\) is a \(2n\times 2n\) matrix over \(\mathbb {Q}\) such that \(M^2=-I_{2n}\), prove that \(M\) is similar to the matrix

State the Noether normalization lemma. Let \(M\) be a maximal ideal of the ring \(A=\mathbb {R}[x,y,z]/(x^2+y^2+z^2+1)\). Determine the field \(A/M\).

Consider the ring homomorphism given by

Is this homomorphism surjective? Determine a minimal generating set for the kernel.