Algebra Qual 2017-03

Problem 1

Let $V=\{a_0+a_1\sqrt[3]{2}+a_2\sqrt[3]{4}\mid a_0,a_1,a_2\in \mathbf{Q}\}\subseteq \mathbf{R}$. This set is a vector space over $\mathbf{Q}$.
(a) Verify $V$ is closed under product (using the usual product operation in $\mathbf{R}$).
(b) Let $T:V\to V$ be the linear transformation defined by $T(v)=(\sqrt[3]{2}+\sqrt[3]{4})v$. Find the matrix that represents $T$ with respect to the basis $\{1,\sqrt[3]{2},\sqrt[3]{4}\}$ for $V$.
(c) Determine the characteristic polynomial for $T$.

Problem 2

Suppose $F$ is a field and $A$ is an $n\times n$ matrix over $F$. Suppose further that $A$ possesses distinct eigenvalues ${\lambda }_1$ and ${\lambda }_2$ with $\dim \mathrm{Null}(A-{\lambda }_1{I}_n)=n-1$. Prove $A$ is diagonalizable.

Problem 3

(a) Suppose $N$ is a normal subgroup of a group $G$ and ${\pi }_N:G\to G/N$ is the usual projection homomorphism, defined by ${\pi }_N(g)=gN$. Prove that if $\varphi :G\to H$ is any homomorphism with $N\le \ker (\varphi )$, then there exists a unique homomorphism $\psi :G/N\to H$ such that $\varphi =\psi \circ {\pi }_N$. (You must explicitly define $\psi$, show it is well defined, show $\varphi =\psi \circ {\pi }_N$, and show that $\psi$ is uniquely determined.)
(b) Prove the:
Third Isomorphism Theorem. If $M,N⊴G$ with $N\le M$, then $(G/N)/(M/N)\cong G/M$.

Problem 4

Explicitly list all group homomorphisms $f:\mathbf{Z}/6\mathbf{Z}\to \mathbf{Z}/12\mathbf{Z}$.

Problem 5

Let $\epsilon :\mathbf{R}[x]\to \mathbf{C}$ be the ring homomorphism that is evaluation at $i$, so $\epsilon (f)=f(i)$. (Here $i$ denotes the complex number sometimes denoted $\sqrt{-1}$.)
(a) Prove that $\ker (\epsilon )=(x^2+1)\subseteq \mathbf{R}[x]$.
(b) Prove that $(x^2+1)$ is a maximal ideal in $\mathbf{R}[x]$.