Homework Assignment 5

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 13.1: 1, 2
• Section 13.2: 3, 4, 5, 7, 10, 12, 13

1. Additional Problem. For $n\ge 1$, we define $\zeta_n \in \mathbb{C}$ to be $\zeta_n = \cos(2\pi/n) + i\sin(2\pi/n)$. (See footnotes¹ for more info.)
   1. Prove that the set of roots in $\mathbb{C}$ of $x^n-1$ is $\{1,\zeta_n,(\zeta_n)^2,\ldots,(\zeta_n)^{n-1}\}$.
   2. Prove that if $p$ is prime, then $\Phi_p(x) = x^{p-1}+x^{p-2}+\cdots+x^2+x+1$ is the minimal polynomial for $\zeta_p$ over $\mathbb{Q}$.
      
      

2. Additional Problem. Let $m,n\in \mathbb{Z}$ with $2\le m<n$. If $p,q\in \mathbb{Z}$ are prime, then $\sqrt[\overset{n}{}]{p} \notin \mathbb{Q}(\sqrt[\overset{m}{}]{q})$.

3. Additional Problem. Let $F\subseteq K \subseteq L$ be fields with $[L:F]$ finite. Then $[K:F] = [L:F]$ if and only if $K = L$.

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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 13.2: 14

2. Additional Problem. In this problem, we’ll find a basis for $\mathbb{Q}(\sqrt[4]{2},\zeta_3)$ over $\mathbb{Q}$. You can freely use the results of the first additional problem to try.
   1. Determine (with proof) the minimal polynomial of $\sqrt[4]{2}$ over $\mathbb{Q}$, and find a basis for $\mathbb{Q}(\sqrt[4]{2})$ over $\mathbb{Q}$.
   2. Prove that $\Phi_3(x)=x^2+x+1$ is the minimal polynomial for $\zeta_3$ over $\mathbb{Q}(\sqrt[4]{2})$, and find a basis for $\mathbb{Q}(\sqrt[4]{2},\zeta_3)$ over $\mathbb{Q}(\sqrt[4]{2})$. (Remember that EIC does NOT apply! Instead, check for roots in $\mathbb{Q}(\sqrt[4]{2})$.)
   3. Find a basis for $\mathbb{Q}(\sqrt[4]{2},\zeta_3)$ over $\mathbb{Q}$.
      
      
3. Additional Problem. Prove that $[\mathbb{Q}(\sqrt[5]{3},\zeta_5):\mathbb{Q}]= 20$.

   First compute $[\mathbb{Q}(\sqrt[5]{3}):\mathbb{Q}]$ and $[\mathbb{Q}(\zeta_5):\mathbb{Q}]$. You can freely use the results of the first additional problem to try.

4. Additional Problem. Suppose $d \in\mathbb{N}$ is not a square (i.e. $\sqrt{d} \notin \mathbb{Z}$). Prove that $[\mathbb{Q}(\sqrt{d},i):\mathbb{Q}]=4$ and that $\mathbb{Q}(\sqrt{d},i) = \mathbb{Q}(\sqrt{d} +i)$.

   There are various approaches to the second part. One way is to directly show both containments, which ultimately rests on showing $\sqrt{d},i \in \mathbb{Q}(\sqrt{d} +i)$. Another approach starts from the observation that it suffices to show $[\mathbb{Q}(\sqrt{d} +i):\mathbb{Q}] = 4$…do you see why? One key hurdle to continue with this approach is to prove that $[\mathbb{Q}(\sqrt{d} +i):\mathbb{Q}] \neq 2$. If $[\mathbb{Q}(\sqrt{d} +i):\mathbb{Q}] = 2$, then every element of $\mathbb{Q}(\sqrt{d} +i)$ can be written as $a+b(\sqrt{d} +i)$ with $a,b\in \mathbb{Q}$; can you find a contradiction?

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Footnotes