Homework Assignment 8

Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

• Section 14.2: 4, 13, 16

1. Additional Problem. Let $p$ and $q$ be distinct primes, and set $L = \mathbb{Q}(\sqrt{p},\sqrt{q})$. Prove that $\operatorname{Aut}(L/\mathbb{Q})$ is the Klein $4$-group and that the proper subfields of $L$ are precisely $\mathbb{Q}$, $\mathbb{Q}(\sqrt{p})$, $\mathbb{Q}(\sqrt{p})$, and $\mathbb{Q}(\sqrt{pq})$, all of which are distinct.

   You can assume that $\sqrt{q} \notin \mathbb{Q}(\sqrt{p})$…or you could (re)prove it for extra practice.

2. Additional Problem. Let $f(x) \in F[x]$, and let $L$ be any extension of $F$ in which $f(x)$ splits. Prove that if $\sigma \in \operatorname{Aut}(L/F)$, then $\sigma$ restricts to an element of $\operatorname{Aut}(F^{f(x)}/F)$ (which, abusing notation, we simply write as $\sigma\in \operatorname{Aut}(F^{f(x)}/F)$). Deduce that if $\gamma$ denotes complex conjugation in $\mathbb{C}$, then for all $f(x) \in \mathbb{Q}[x]$, $\gamma \in \operatorname{Aut}(\mathbb{Q}^{f(x)}/\mathbb{Q})$. (See the footnote¹ for notation.)

3. Additional Problem. Let $f(x) \in \mathbb{Q}[x]$. Suppose that $f(x)$ has exactly two roots in $\mathbb{C}$ that are not in $\mathbb{R}$. Prove that, when viewing $\operatorname{Aut}(\mathbb{Q}^{f(x)}/\mathbb{Q})$ as permutations of the roots of $f(x)$, $\operatorname{Aut}(\mathbb{Q}^{f(x)}/\mathbb{Q})$ contains a transposition.

4. Additional Problem. Suppose $K/F$ is a Galois extension, and let $G= \operatorname{Aut}(K/F)$. For all $a\in K$, define the “norm” and “trace” maps as $\operatorname{N}(a) = \prod_{\sigma\in G} \sigma(a)$ and $\operatorname{Tr}(a) = \prod_{\sigma\in G} \sigma(a)$. Prove that for all $a\in K$, $\operatorname{N}(a)$ and $\operatorname{Tr}(a)$ are in $F$.

   Show that $\operatorname{N}(a)$ and $\operatorname{Tr}(a)$ are in the fixed field of $G$.

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Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Section 14.2: 3

   You can use the Additional Problem above about “bi-quadratic” extensions.

2. Section 14.2: 11

   Please give a small proof when answering the final question (which is independent from the first part of the problem).

3. Section 14.2: 14

   Note that $(\sqrt{2+\sqrt{2}})(\sqrt{2-\sqrt{2}}) = \sqrt{2}$.

4. Additional Problem. Let $f(x) \in \mathbb{Q}[x]$ be irreducible. Suppose that $\deg f(x)=p$ is prime and $f(x)$ has exactly two roots in $\mathbb{C}$ that are not in $\mathbb{R}$. Prove that the Galois group of $f(x)$ over $\mathbb{Q}$ is isomorphic to $\operatorname{Sym}(p)$.

   Make use of an Additional Problem above and Exercise #5 from Section 3.5. (You may also want to try to prove Exercise #5 from Section 3.5 using Exercise #4, which was a “Problem to Try” from long ago.)

Footnotes