Algebra Qual 2021-09

Problem 1

Let $n$ be a number between $0$ and $10$. Compute $n^{111}(\mathrm{mod}11)$, expressing your answer as a number between $0$ and $10$. Give as detailed a proof as you can, justifying every step, no matter who trivial you think it is.

Problem 2

Let $G$ be a group of order $2n$ for some positive integer $n>1$.
(a) Prove there exists a subgroup $K$ of $G$ of order $2$.
(b) Suppose $K$ in (a) is a normal subgroup. Prove that $K$ is contained in the center $\mathrm{Z}(G)$. (Recall $\mathrm{Z}(G)=\{a\in G\mid ab=ba\text{ for all }b\in G\}$.)

Problem 3

Consider the additive group of integers $\mathbf{Z}$.
(a) Prove that every subgroup of $\mathbf{Z}$ is a cyclic group.
(b) Prove that every homomorphic image of $\mathbf{Z}$ is a cyclic group.
(c) Now consider the ring $\mathbf{Z}$. Exhibit a prime ideal of $\mathbf{Z}$ that is not maximal.

Problem 4

Let $i\in \mathbf{C}$ be the usual root of unity, with $i^2=-1$, and let $\mathbf{Z}[i]=\{a+bi\mid a,b\in \mathbf{Z}\}$ be the ring of Gaussian integers.
(a) Prove that there exists a (nonzero) ring homomorphism $\mathbf{Z}[i]\to {\mathbf{Z}}_5$.
(b) Compute the kernel of your homomorphism explicitly, and state the conclusion given by the First Isomorphism Theorem.

Problem 5

Let $a$ and $b$ be real numbers and let $A\in {\mathbf{R}}^{3\times 3}$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$.
(a) Determine all eigenvalues and representative eigenvectors of $A$ together with their algebraic multiplicities. (Hint: $A=(a-b)I+bJ$ where $J$ is the $3\times 3$ matrix each of whose entries equals $1$.)
(b) Is $A$ diagonalizable? Justify your answer.
(c) Determine the minimal polynomial of $A$.