Algebra Qual 2023-06

Problem 1

Let $G$ denote the set of invertible $2\times 2$ matrices with values in a field. Prove $G$ is a group by defining a group law, identity element, and verifying the axioms. Credit is based on completeness.

Problem 2

Let $G$ be a finite group. Prove from the definitions that there exists a number $N$ such that $a^N=e$ for all $a\in G$.m 2

Problem 3

Suppose $R$ is a PID (principal ideal domain). Prove that an ideal $I\subset R$ is maximal if and only if $I=⟨p⟩$ for a prime $p\in R$. (By definition, an element $p$ is prime if whenever $p\mid ab$ then $p\mid a$ or $p\mid b$. If you use the fact that prime implies irreducible, you have to prove it.)

Problem 4

Let $\mathcal{C}([0,1])$ be the (commutative) ring of continuous, real-valued functions on the unit interval, and let

Prove that $M$ is a maximal ideal.

Problem 5

Suppose $V$ is a vector space, and ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n$ are in $V$. Prove that either ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$ are linearly independent, or there exists a number $k\le n$ such that ${\mathbf{v}}_k$ is a linear combination of ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_{k-1}$.