Algebra II - May 2021

Problem 1.

Let $p$ be an odd prime number, and let $\zeta =e^{\frac{2\pi i}{p}}\in \mathbb{C}$ be a primitive $p$th root of unity. Let $G=\mathrm{Gal}(\mathbb{Q}(\zeta )/\mathbb{Q})$ and recall the isomorphism $\chi :G\to (\mathbb{Z}/p\mathbb{Z}{)}^{\times }$ determined by $\sigma (\zeta )={\zeta }^{\chi (\sigma )}$ for $\sigma \in G$. For $i\in (\mathbb{Z}/p\mathbb{Z}{)}^{\times }$, let ${\sigma }_i={\chi }^{-1}(i)\in G$.

Let $H\subset G$ be the index two subgroup and let $S\subset (\mathbb{Z}/p\mathbb{Z}{)}^{\times }$ be the set of squares, so that $\chi (H)=S$. Let $N=(\mathbb{Z}/p\mathbb{Z}{)}^{\times }-S$ be the set of non-squares. Let $\alpha \in \mathbb{Q}(\zeta )$ be the element

and let $\beta \in \mathbb{Q}(\zeta )$ be the element

1. Show that $\alpha +\beta =-1$ and that $\alpha$ and $\beta$ are fixed by $H$.
2. Show that, if $\tau \in G-H$, then $\tau (\alpha )=\beta$ and $\tau (\beta )=\alpha$. Deduce that $\alpha \beta \in \mathbb{Q}$.
3. Show that $$\alpha \beta =\sum _{j\in N}\sum _{i\in S}{\zeta }^{i+ij}.$$
4. Use the previous formula to compute which rational number $\alpha \beta$ is. (Hint: the answer will depend on whether $-1\in S$ or $-1\in N$.)
5. Note that $[\mathbb{Q}(\zeta {)}^H:\mathbb{Q}]=2$, so $\mathbb{Q}(\zeta {)}^H=\mathbb{Q}(\sqrt{d})$ for some squarefree $d$. What is $d$?

Problem 2.

Let $p$ be a prime number and let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $p$. Let $F_f$ be the splitting field of $f$ in $\mathbb{C}$. Suppose that $f(x)$ has $p-2$ real roots and 2 non-real roots in $\mathbb{C}$. Show that the Galois group $\mathrm{Gal}(F_f/\mathbb{Q})$ is $S_p$.

Problem 3.

Let $M$ be an $R$-module and assume that there is an isomorphism $\phi :R\to M{\otimes }_{R}N$ for some $R$-module $N$.

1. Show that $M$ is finitely generated (Hint: write $\phi (1)=\sum _{i=1}^n{x}_i\otimes y_i$ with $x_i\in M$ and $y_i\in N$ and show that $x_1,\dots ,x_n$ generate $M$ as an $R$-module.)
2. Show that $M_{\mathfrak{m}}\cong R_{\mathfrak{m}}$ as $R_{\mathfrak{m}}$-modules for every maximal ideal $\mathfrak{m}\subset R$.

Problem 4.

Assume that $R$ is an integral domain, and let $M$ be an injective $R$-module.

1. Show that $M$ is divisible. That is, show that for any $m\in M$ and any $r\in R$, there is an $m_0\in M$ such that $m=r{m}_0$.
2. Now suppose that $M$ is non-zero and finitely generated. Show that $R$ is a field. (Hint: show that $M_{\mathfrak{m}}=0$ for all non-zero maximal ideals $\mathfrak{m}\subset R$.)