Algebra II - August 2024

Problem 1.

1. Show that the quotient group $\mathbb{Q}/\mathbb{Z}$ is an injective object in the category of abelian groups.
2. Show that there are no nonzero projective objects in the category of torsion abelian groups. Hint: Use the previous part of the problem

Problem 2.

Let $G$ be an infinite cyclic group.

1. Show that $\mathbb{Z}[G]$ is isomorphic to the ring $\mathbb{Z}[x,x^{-1}]$.
2. Show that the sequence
   $$0\to \mathbb{Z}[x,x^{-1}]\stackrel{\delta }{\to }\mathbb{Z}[x,x^{-1}]\stackrel{\epsilon }{\to }\mathbb{Z}\to 0,$$
   where $\delta (f(x))=(x-1)f(x)$ and $\epsilon (f(x))=f(1)$, is a free resolution of $\mathbb{Z}$.
3. Let $M$ be a $G$-module. Show that
   1. $H^0(G,M)=M^G$, where $M^G$ denotes the $G$-invariant submodule of $M$,
   2. $H^1(G,M)\cong M/(g-1)M$, where $g$ is a generator of $G$,
   3. $H^i(G,M)=0$ for $i\ge 2$.

Problem 3.

Let $p$ be a prime and ${\zeta }_p$ a primitive $p$-th root of unity. For a positive integer $d$, show that the field $\mathbb{Q}({\zeta }_p)$ has a subfield $F$ such that $[F:\mathbb{Q}]=d$ if and only if $p\equiv 1\mathrm{mod}d$.

Problem 4.

Let $p$ be a prime number and ${\mathbb{F}}_p$ the finite field with $p$ elements. Let $K={\mathbb{F}}_p(t)$ be the field of rational functions in the variable $t$ over ${\mathbb{F}}_p$.

1. Show that the splitting field of the polynomial $x^p-t\in K[x]$ over $K$ is inseparable.
2. Show that the splitting field of $x^p-t-1\in K[x]$ over $K$ is the same as the splitting field of $x^p-t$.
3. Show that $K$ has a single degree $p$ inseparable extension. Hint: first show that an inseparable degree $p$ extension of $K$ must be of the form $K(\alpha )$ for some inseparable element $\alpha$ and use what you know about the form that inseparable polynomials must have.

Problem 5.

Let $p$ be a prime and $G$ the multiplicative group of of $3\times 3$ matrices over ${\mathbb{F}}_p$ which are upper triangular and have 1s on the diagonal. Let $V$ be the space of complex functions on ${\mathbb{F}}_p$.

For any $z\in \mathbb{C}$ such that $z^p=1$, define a representation $R_z$ of the group $G$ on $V$ by

1. Show that $R_z$ is a representation and compute $R_z(g)$ for all $g\in G$.
2. Show that $R_z$ is irreducible if and only if $z\ne 1$.
3. Classify all 1-dimensional representations of $G$ and show that $R_1$ decomposes into a direct sum of 1-dimensional representations, where each occurs exactly once. Hint: For the first part, consider the action of $R_z\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right)$ on eigenvectors
4. Use the previous parts to classify all irreducible representations of $G$.