Algebra I - August 2025

August 2025 Algebra I Qualifying Exam

Problem 1.

Let $F$ be an algebraically closed field, $V$ a nonzero finite-dimensional vector space over $F$, and $T:V\to V$ a linear transformation.

1. Show that $T$ is diagonalizable iff its minimal polynomial has no repeated roots. Hint: Use the Structure Theorem for finitely generated $F[t]$-modules.

2. Show that $T$ is nilpotent iff its minimal polynomial is $t^k$, for some $k\in\mathbb N$.

Problem 2.

Let $SL_2(\mathbb F_p)$ be the group of $2\times2$ matrices with entries in the finite field $\mathbb F_p$ and determinant equal to 1.

1. Find the size of $SL_2(\mathbb F_p)$.

2. Find a $p$-Sylow subgroup of $SL_2(\mathbb F_p)$.

3. How many $p$-Sylow subgroups are there in $SL_2(\mathbb F_p)$?

Problem 3.

Let $\{1\}\xrightarrow{i}K\to G\xrightarrow{\pi}H\to\{1\}$ be a short exact sequence of finite groups. Show that if $P\subset G$ is a $p$-Sylow subgroup of $G$, then $\pi(P)$ is a $p$-Sylow subgroup of $H$.

Problem 4.

Let $R$ be a commutative ring, and let $f\in R$ be an element.

1. Show that the localization $R_f$ is isomorphic to $R[x]/(1-xf)$ as a ring.

2. Show that if $f=g+n$, where $n$ is nilpotent, then $R_f$ is isomorphic to $R_g$.

Problem 5.

Consider $\mathbb Q$ as a $\mathbb Z$-module.

1. State the definition of Noetherian $R$-module. Is $\mathbb Q$ Noetherian? Justify your answer.

2. State the definition of a projective $R$-module. Is $\mathbb Q$ projective? Justify your answer.

Problem 6.

Let $R$ be a commutative ring, and let $P\subset R$ be a prime ideal.

1. Show that if $I,J\subset R$ are ideals such that $I\cap J=P$, then $I=P$ or $J=P$.

2. Show that $R/P$ is an indecomposable $R$-module, meaning it is not isomorphic to a direct sum of (nonzero) $R$-modules.