Algebra Qual 2022-01

Problem 1

Let $\sigma :G\to H$ be a group epimorphism. Let $N$ be a normal subgroup of $G$ and $K=\sigma (N)$, the image of $N$ in $H$.
(a) Prove that $K$ is a normal subgroup of $H$. Give an example to show that this is not true if $\sigma$ is not onto.
(b) Under what conditions does $\sigma$ induce a homomorphism $G/N\to H/K$, and when is this an isomorphism? Prove your answer.

Problem 2

The dihedral group, $D_8$, is the group of eight rigid symmetries of a square. Prove that $D_8$ is not the internal direct product of two of its proper subgroups.

Problem 3

Let $A$ be a commutative ring with $1$. The dimension of $A$ is the maximum length $d$ of a chain of prime ideals ${\mathfrak{p}}_0⊊{\mathfrak{p}}_1⊊\cdots ⊊{\mathfrak{p}}_d$. Prove that if $A$ is a PID, the dimension of $A$ is at most 1.

Problem 4

Let ${\mathbf{Z}}_2=\{0,1\}$ be the field of two elements. The quotient ring ${\mathbf{Z}}_2[x]/⟨x^3+x+1⟩$ is a field of cardinality 8, containing ${\mathbf{Z}}_2$. Let $\pi :{\mathbf{Z}}_2[x]\to {\mathbf{Z}}_2[x]/⟨x^3+x+1⟩$ be the natural projection.
(a) Write down a set of eight distinct coset representatives for the elements of this field.
(b) Determine the multiplicative inverse of $\pi (x)$ in terms of your coset representatives.

Problem 5

Let $T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ be the orthogonal projection to a $1$-dimensional linear subspace $L\subset {\mathbf{R}}^3$.
(a) List the eigenvalues of $T$.
(b) Write the characteristic polynomial $p_T(x)$ for $T$.
(c) Is $T$ diagonalizable? Justify your answer.