Algebra Qual 2021-06

Problem 1

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

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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 $x \in A_{5}$ .

Prove that the function $γ : A_{5} \to Conj (A_{5})$ , taking $s \in A_{5}$ to $γ_{s}$ , is a surjective homomorphism.
Prove that $A_{5}$ is isomorphic to $Conj (A_{5})$ .

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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 $\operatorname{Aut}(A_5)$. The group of {\bfseries conjugations} of $A_5$, denoted $\operatorname{Conj}(A_5)$, is the subgroup of $\operatorname{Aut}(A_5)$ consisting of automorphisms of the form $\gamma_s:=s(-)s^{-1}$ where $s\in A_5$. Explicitly, $\gamma_s(x)=sxs^{-1}$ for any $x\in A_5$.
\begin{enumerate}[label=\alph*)]
	\item Prove that the function $\gamma:A_5\to \operatorname{Conj}(A_5)$, taking $s\in A_5$ to $\gamma_s$, is a surjective homomorphism.
	\item Prove that $A_5$ is isomorphic to $\operatorname{Conj}(A_5)$.
\end{enumerate}

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

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

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

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Let ${\bf Z}[X]$ be the ring of polynmomials with integer coefficients, and let $K\subset {\bf Z}[X]$ be the kernel of the ``evaluation at $1

Problem 4

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

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

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Let $A=\begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Determine whether $A$ is diagonalizable, and if so, give its diagonal form along with a diagonalizing matrix.
	\item Compute $A^{42}$. Remember to show all work.
\end{enumerate}

Problem 5

Let $F_{9}$ denote the field of nine elements.

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

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Let ${\bf F}_9$ denote the field of nine elements.
\begin{enumerate}[label=\alph*)]
	\item Show that each nonzero $a\in {\bf F}_9$ is a root of $X^3-1=(X-1)(X^2+1)(X^4+1)\in {\bf F}_3[X]$.
	\item Use the Pigeonhole Principle to prove that ${\bf F}_9$ has an element of multiplicative order 8. (Include a proof that the Pigeonhole Principle applies.)
\end{enumerate}

' homomorphism > \begin{align*} > \varepsilon_1:{\bf Z}[X] &\to {\bf Z}_3\\ > f(X) &\mapsto [f(1)]_3. > \end{align*} > \begin{enumerate}[label=\alph*)] > \item Characterize $K$ as a set. > \item Determine whether $K$ is a maximal ideal. Fully justify your conclusion. > \item Determine whether $K$ is a principal ideal. Justify by either exhibiting a generator or proving that there isn't one. > \end{enumerate} > ```

Problem 4

!Diagonalization and matrix powers

Problem 5

!The field with nine elements