Homework Assignment 7

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 14.1: 4, 5

1. Additional Problem (Variant of Exercise 1(a), Section 14.1). Let $L$ be any field. Suppose $F$ is a subfield, and $\alpha_1,\ldots,\alpha_n \in L$.
   1. Let $\sigma \in \operatorname{Aut}(L/F)$. Prove that if $\sigma$ fixes each $\alpha_i$, then $\sigma$ fixes $F(\alpha_1,\ldots,\alpha_n)$.
   2. Let $\sigma,\tau \in \operatorname{Aut}(L/F)$. Prove that if $\sigma(\alpha_i) = \tau(\alpha_i)$ for all $i$, then $\sigma = \tau$.

   For part a, consider the fixed field of $\sigma$. (See page 560 for the defintion of the fixed field.)

   For part b, try showing $\tau^{-1}\sigma = 1$. The conclusion of part b is usually stated as “$\sigma$ is determined by its action on the generators.”

2. Additional Problem (Special case of Theorem 8, Section 13.1). Let $p(x) \in F[x]$. Suppose $\alpha$ and $\beta$ are roots of $p(x)$ (in some field in which $p(x)$ splits). Then there exists an isomorphism from $F(\alpha)$ to $F(\beta)$ that fixes $F$ and sends $\alpha$ to $\beta$. Moreover, there is a unique such isomorphism with these properties.

   The intention is that you simply explain how to use Theorem 8 from Section 13.1 to prove this result; however, make sure to briefly address the “uniqueness” portion too, for which the previous Additional Problem should help.

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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. Additional Problem. Consider the extension $\mathbb{Q}(\zeta_5)/\mathbb{Q}$.
   1. Describe, with proof, the elements in $\operatorname{Aut}(\mathbb{Q}(\zeta_5)/\mathbb{Q})$ by specifying where each one sends $\zeta_5$.
   2. Prove that $\operatorname{Aut}(\mathbb{Q}(\zeta_5)/\mathbb{Q})\cong \mathbb{Z}/4\mathbb{Z}$.

   When determining the elements of $\operatorname{Aut}(\mathbb{Q}(\zeta_5)/\mathbb{Q})$, you will probably build a list of potential automorphisms first. Please make sure to justify why each one is or is not actually an automorphism. One approach is to use Theorem 8 from Section 13.1 (see the Additional Problem above), but there are other approaches.

2. Additional Problem. Let $p$ be a prime and $n\ge 1$. Define $\phi_p:\mathbb{F}_{p^n} \rightarrow \mathbb{F}_{p^n}:a\mapsto a^p$.
   1. Prove that $\phi_p \in \operatorname{Aut}(\mathbb{F}_{p^n}/\mathbb{F}_{p})$.
   2. Prove that if $1\le k \le n-1$, then $\phi_p^k \neq 1$ (where $\phi_p^k$ means $\phi_p$ composed with itself $k$ times).
   3. Prove that $\operatorname{Aut}(\mathbb{F}_{p^n}/\mathbb{F}_{p})$ cyclic of order $n$ and generated by $\phi_p$.

   Recall that $\mathbb{F}_{p^n}$ is the splitting field for $x^{p^n} - x$ over $\mathbb{F}_{p}$, and we proved that $\mathbb{F}_{p^n}$ is a field with $p^n$ elements (i.e. that $[\mathbb{F}_{p^n}:\mathbb{F}_{p}] = n$).

   For part b, write out a formula for $\phi_p^k$. To show $\phi_p^k \neq 1$, you need to show that $\phi_p^k(a) \neq a$ for some $a\in \mathbb{F}_{p^n}$; in other words, show that $\phi_p^k(x) - x = 0$ does NOT have $p^n$ solutions.

   For part c, note that Proposition 5 from Section 14.1 applies.

   Note: There is an example in the book that talks through this problem. I encourage you to avoid looking at the example until you’ve taken a go yourself. Regardless, please write up your solution in your own words with full details. Let me know if you have questions.