Algebra Qual 2021-06

Problem 1

Prove from the definition along that there are no nonabelian groups of order less than $5$ .

Problem 2

Let $A_{5}$ denote the alternating group on a $5$ -element set ${1, 2, 3, 4, 5}$ . The set of automorphisms of $A_{5}$ form a group, denoted $Aut (A_{5})$ . The group of conjugations of $A_{5}$ , denoted $Conj (A_{5})$ , is the subgroup of $Aut (A_{5})$ consisting of automorphisms of the form $γ_{s} := s (-) s^{- 1}$ where $s \in A_{5}$ . Explicitly, $γ_{s} (x) = s x s^{- 1}$ for any $x \in A_{5}$ .
(a) Prove that the function $γ : A_{5} \to Conj (A_{5})$ , taking $s \in A_{5}$ to $γ_{s}$ , is a surjective homomorphism.
(b) Prove that $A_{5}$ is isomorphic to $Conj (A_{5})$ .

Problem 3

Let $Z [X]$ be the ring of polynomials with integer coefficients, and let $K \subset Z [X]$ be the kernel of the "evaluation at $1$ " homomorphism

\begin{aligned} ε_{1} : Z [X] & \to Z_{3} \\ f (X) & \mapsto [f (1)]_{3} . \end{aligned}

(a) Characterize $K$ as a set.
(b) Determine whether $K$ is a maximal ideal. Fully justify your conclusion.
(c) Determine whether $K$ is a principal ideal. Justify by either exhibiting a generator or proving that there isn't one.

Problem 4

Let $A = [\begin{matrix} 0 & 0 & - 2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{matrix}]$ .

(a) Determine whether $A$ is diagonalizable, and if so, give its diagonal form along with a diagonalizing matrix.
(b) Compute $A^{42}$ . Remember to show all work.

Problem 5

Let $F_{9}$ denote the field of nine elements.
(a) Show that each nonzero $a \in F_{9}$ is a root of $X^{8} - 1 = (X - 1) (X + 1) (X^{2} + 1) (X^{4} + 1) \in F_{3} [X]$ .
(b) Use the Pigeonhole Principle to prove that $F_{9}$ has an element of multiplicative order 8. (Include a proof that the Pigeonhole Principle applies.)