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.

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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 $a \in G$ .

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

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 $p ∣ a b$ then $p ∣ a$ or $p ∣ b$ . If you use the fact that prime implies irreducible, you have to prove it.)

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Suppose $R$ is a PID (principal ideal domain). Prove that an ideal $I\subset R$ is maximal if and only if $I=\langle p\rangle$ for a prime $p\in R$. (By definition, an element $p$ is {\bfseries 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 $C ([0, 1])$ be the (commutative) ring of continuous, real-valued functions on the unit interval, and let

$M = {f \in C ([0, 1]) ∣ f (\frac{1}{2}) = 0} .$

Prove that $M$ is a maximal ideal.

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Let $\mathcal{C}([0,1])$ be the (commutative) ring of continuous, real-valued functions on the unit interval, and let
\[
	M=\left\{f\in \mathcal{C}([0,1])\,\mid\, f\left(\frac{1}{2}\right)=0\right\}.
\]
Prove that $M$ is a maximal ideal.

Problem 5

Suppose $V$ is a vector space, and $v_{1}, v_{2}, \dots, v_{n}$ are in $V$ . Prove that either $v_{1}, \dots, v_{n}$ are linearly independent, or there exists a number $k \leq n$ such that $v_{k}$ is a linear combination of $v_{1}, \dots, v_{k - 1}$ .

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