Algebra II - August 2025

Problem 1.

Let $p$ be a prime, and let ${\zeta }_n=e^{2\pi i/n}$ for $n\in \mathbb{N}$.

1. Describe the Galois group of the extension $\mathbb{Q}\subset \mathbb{Q}({\zeta }_p)$. For which values of $p$ does this Galois group have a quotient isomorphic to $\mathbb{Z}/5\mathbb{Z}$?
2. Show that there exists an extension of $\mathbb{Q}$ with Galois group isomorphic to $\mathbb{Z}/5\mathbb{Z}\times \mathbb{Z}/5\mathbb{Z}$.

Problem 2.

Let $F\subset K$ be a Galois extension with Galois group $G$. Recall that the trace map $T:K\to K$ can be defined as

1. Show that $T$ is an $F$-linear map, and it takes values in $F$, so we can write $T:K\to F$.
2. Using independence of characters, show that there exists $c\in K$ such that $T(c)\ne 0$.
3. Assume that $G\simeq C_n$, generated by an element $g\in G$. Show that if $T(\alpha )=0$, then
   $$\beta :=\frac{1}{T(c)}(\alpha g(c)+(\alpha +g(\alpha ))g^2(c)+\cdots +(\alpha +g(\alpha )+\cdots +g^{n-2}(\alpha ))g^{n-1}(c))$$
   satisfies the property that $\alpha =\beta -g(\beta )$.

Problem 3.

Let $F$ be a field of characteristic $p>0$, $c\in F$, and consider the polynomial

1. Show that if $f(x)$ is irreducible, then $K=F[x]/(x^p-x+c)$ is a Galois extension of $F$ with Galois group isomorphic to $C_p$.
2. Show conversely that if $F\subset K$ is a Galois extension with Galois group isomorphic to $C_p$, then it must be isomorphic to the example in part (a). Hint: Observe that $-1\in K$ has trace 0, and then use the previous problem.

Problem 4.

Describe all abelian groups $A$ which can occur in a short exact sequence

Problem 5.

Describe all finite-dimensional complex irreducible representations of $D_4$, the dihedral group of order 8.

Problem 6.

Let $R=F[x,y]$, and let $I\subset R$ be the ideal generated by $x$ and $y$.

1. Compute ${\mathrm{Ext}}_R^i(I,R/I)$ for all $i\ge 0$, using a projective resolution.
2. Show that ${\mathrm{Ext}}_R^1(I,R/I)\simeq {\mathrm{Ext}}_R^2(R/I,R/I)$, using a long exact sequence.