Algebra II - May 2022

Instructions

You will be graded on 4 out of 5 questions. If you submit solutions to all 5 please specify on the first page which question you choose to be ignored.

Problem 1.

Let $\alpha =\sqrt{2}+\sqrt{5}$.

1. Find the minimal polynomial $f$ of $\alpha$ over $\mathbb{Q}$.
2. Show that $\mathbb{Q}(\alpha )=\mathbb{Q}(\sqrt{2},\sqrt{5})$ and hence that $\mathbb{Q}(\alpha )$ is the splitting field of $f$ over $\mathbb{Q}$.
3. Find the other roots of $f$, and write them in terms of $\alpha$.

Problem 2.

Explicitly find all the intermediate fields of the extension $\mathbb{Q}({\xi }_8)/\mathbb{Q}$ where ${\xi }_8$ is a primitive 8th root of unity.

Problem 3.

Show directly (i.e. without appealing to the fact that $A$ is injective) that if $A$ is a divisible $\mathbb{Z}$-module, and $M$ is any $\mathbb{Z}$-module, then a short exact sequence

splits.

Problem 4.

Let $R$ be the ring $\mathbb{C}[x]/(x^n)$ for some $n\ge 2$, and view $\mathbb{C}$ as an $R$-module via the quotient $\mathbb{C}\cong R/(x)$.

1. Show that
   $$\cdots \stackrel{f\mapsto x^{n-1}f}{\to }\mathbb{C}[x]/(x^n)\stackrel{f\mapsto xf}{\to }\mathbb{C}[x]/(x^n)\stackrel{f\mapsto x^{n-1}f}{\to }\mathbb{C}[x]/(x^n)\stackrel{f\mapsto xf}{\to }\mathbb{C}[x]/(x^n)\stackrel{/(x)}{\to }\mathbb{C}$$
   is a projective resolution of $\mathbb{C}$ as an $R$-module, where $\dots$ indicates the the sequence continues infinitely in the same manner.
2. For all $i\ge 0$, compute the dimension of ${\mathrm{Tor}}_i^R(\mathbb{C},\mathbb{C}[x]/(x^2))$ as a complex vector space.

Problem 5.

Show that if $R$ is a semi-simple ring, and $M$ is a simple $R$-module, then $R$ has a submodule isomorphic to $M$.