Algebra Qual 2014-09

Problem 1

Let $G$ be a finite abelian group of odd order. Let $\varphi :G\to G$ be the function defined by $\varphi (g)=g^2$ for all $g\in G$. Prove that $\varphi$ is an automorphism.

Problem 2

Let $G$ be a group and suppose $H\le G$. The normalizer of $H$ in $G$ is defined to be $N(H)=\{g\in G|gH=Hg\}$ and the centralizer of $H$ in $G$ is defined to be $C(H)=\{g\in G|gh=hg\text{ for all }h\in H\}$.
(a) Prove that $N(H)$ is a subgroup of $G$.
(b) Prove that $C(H)$ is a normal subgroup of $N(H)$ and that $N(H)/C(H)$ is isomorphic to a subgroup of $\mathrm{Aut}(H)$.

Problem 3

Let $V$ denote the real vector space of polynomials in $x$ of degree at most 3. Let $\mathcal{B}=\{1,x,x^2,x^3\}$ be a basis for $V$ and $T:V\to V$ be the function defined by $T(f(x))=f(x)+f^{\prime }(x)$.
(a) Prove that $T$ is a linear transformation.
(b) Find $[T{]}_{\mathcal{B}}$, the matrix representation for $T$ in terms of the basis $\mathcal{B}$.
(c) Is $T$ diagonalizable? If yes, find a matrix $A$ so that $A[T{]}_{\mathcal{B}}{A}^{-1}$ is diagonal, otherwise explain why $T$ is not diagonalizable.

Problem 4

Let $f(x)=x^3+x+1\in {\mathbf{Z}}_5[x]$.
(a) Prove that $f(x)$ is irreducible.
(b) Prove that $⟨f(x)⟩$ is a maximal ideal.
(c) What is the cardinality of ${\mathbf{Z}}_5[x]/⟨f(x)⟩$? Justify.

Problem 5

Let $R$ be a commutative ring. The nilradical of $R$ is defined to be $N=\{r\in R|r^n=0\text{ for some }n\in \mathbf{N}\}$.
(a) Prove that $N$ is an ideal of $R$.
(b) Prove that $N$ is contained in the intersection of all prime ideals of $R$.