Math 6310 • Algebra

Let \(F\) be a field with \(\mathbb{Q}\subset F\subset\mathbb{C}\), where \(F/\mathbb{Q}\) is a
finite Galois extension. Let \(\alpha\in F\) and let \(f(X)\in\mathbb{Q}[X]\) be its minimal monic
polynomial. Assume that the absolute value \(|\alpha|\) is 1, and that \(\operatorname{Gal}(F/\mathbb{Q})\)
is abelian.

\begin{enumerate}
\item[(a)]
Show that \(F\) is closed under complex conjugation.
\item[(b)]
Prove that \(|\beta|=1\) for every complex root \(\beta\) of \(f(X)\).
\item[(c)]
Writing \(f(X)=X^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0\), show that \(|a_i|\le 2^n\) for all \(i\)
with \(0\le i<n\).
\item[(d)]
Prove that \(F\) contains only finitely many algebraic integers having absolute value 1 and
deduce that each of these is a root of unity.
\end{enumerate}