\documentclass{article}
\usepackage{amsfonts,amsmath,amssymb}
\usepackage{tikz}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\usepackage{enumerate}
\begin{document}
\section*{Math 403/503 Spring 2024}
\subsection*{Homework 8, due April 10}
\begin{enumerate}
\item Show that each of the following numbers is algebraic over $\Q$.
\begin{enumerate}[{\rm (a)}]
\item $\alpha=\sqrt[3]{\sqrt{1/3}-5}$
\item $\beta=\sqrt{2}-\sqrt[3]{5}$
\end{enumerate}
\item Let $E/F$ be a field extension. Prove that if $\alpha\in E$ is transcendental over $F$, then $\alpha^k$ is transcendental over $F$ for any positive integer $k$.
\item Let $E/F$ be a field extension and let $F_1$ and $F_2$ be subfields of $E$ containing $F$. Let $d_i=[E:F_i]$ for $i=1,2$. If $d_1$ and $d_2$ are relatively prime, show that $[E:F]$ is at least $d_1d_2$.
\[\begin{tikzpicture}
  \node (E) at (0,0) {$E$};
  \node (F1) at (-1,-1) {$F_1$};
  \node (F2) at (1,-1) {$F_2$};
  \node (F) at (0,-2) {$F$};
  \draw (E) -- (F1) -- (F);
  \draw (E) -- (F2) -- (F);
\end{tikzpicture}\]
\item Find the minimal polynomial of the number over $\Q$:
\begin{enumerate}[{\rm (a)}]
\item $\gamma=1+\sqrt[3]{2}$
\item $z=\cos\theta+i\sin\theta$ for $\theta=2\pi/p$ with $p$ prime. (\emph{Hint:} Consider $z^p$; use Euler's formula.)
\end{enumerate}
\item Find a basis for the field extension. What is the degree of the extension?
\begin{enumerate}[{\rm (a)}]
\item $\Q(\sqrt{8})$ over $\Q(\sqrt{2})$
\item $\Q(\sqrt{3},\sqrt{5},\sqrt{7})$ over $\Q$
\end{enumerate}
\item Consider the field extension $\Q(\sqrt[4]{3}, i)$ over $\Q$.
\begin{enumerate}[{\rm (a)}]
\item Find a basis for the field extension $\Q(\sqrt[4]{3},i)$ over $\Q$. Conclude that $[\Q(\sqrt[4]{3},i) : \Q] = 8$.
\item Find all subfields $F$ of $\Q(\sqrt[4]{3},i)$ such that $[F : \Q] = 2$.
\item Find all subfields $F$ of $\Q(\sqrt[4]{3},i)$ such that $[F : \Q] = 4$.
\end{enumerate}
\end{enumerate}
\end{document}
