books/napkin/discriminant.tex

\chapter{More properties of the discriminant}
I'll remind you that the discriminant of a number field $K$ is given by
\[
\Delta_K \defeq \det
\begin{bmatrix}
	\sigma_1(\alpha_1) & \dots & \sigma_n(\alpha_1) \\
	\vdots & \ddots & \vdots \\
	\sigma_1(\alpha_n) & \dots & \sigma_n(\alpha_n) \\
\end{bmatrix}^2
\]
where $\alpha_1$, \dots, $\alpha_n$ is a $\ZZ$-basis for $K$,
and the $\sigma_i$ are the $n$ embeddings of $K$ into $\CC$.

Several examples, properties, and equivalent definitions follow.

\section\problemhead
\begin{sproblem}[Discriminant of cyclotomic field]
	\label{prob:discrim_cyclotomic_field}
	Let $p$ be an odd rational prime and $\zeta_p$ a primitive $p$th root of unity.
	Let $K = \QQ(\zeta_p)$.
	Show that \[ \Delta_K = (-1)^{\frac{p-1}{2}} p^{p-2}. \]
	\begin{hint}
		Direct linear algebra computation.
	\end{hint}
\end{sproblem}

\begin{sproblem}[Trace representation of $\Delta_K$]
	\gim
	Let $\alpha_1$, \dots, $\alpha_n$ be a basis for $\OO_K$.
	Prove that
	\[ \Delta_K
		=
		\det
		\begin{bmatrix}
			\TrK(\alpha_1^2) & \TrK(\alpha_1\alpha_2) & \dots & \TrK(\alpha_1\alpha_n) \\
			\TrK(\alpha_2\alpha_1) & \TrK(\alpha_2^2) & \dots & \TrK(\alpha_2\alpha_n) \\
			\qquad\vdots & \qquad\vdots & \ddots & \qquad\vdots \\
			\TrK(\alpha_n\alpha_1) & \TrK(\alpha_n\alpha_2) & \dots & \TrK(\alpha_n\alpha_n) \\
		\end{bmatrix}.
	\]
	In particular, $\Delta_K$ is an integer.
	\label{prob:trace_discriminant}
	\begin{hint}
		Let $M$ be the ``embedding'' matrix.
		Look at $M^\top M$, where $M^\top$ is the transpose matrix.
	\end{hint}
\end{sproblem}


\begin{sproblem}[Root representation of $\Delta_K$]
	The \vocab{discriminant} of a quadratic polynomial $Ax^2+Bx+C$ is defined as $B^2-4AC$.
	More generally, the polynomial discriminant of a polynomial $f \in \ZZ[x]$ of degree $n$ is
	\[ \Delta(f) \defeq c^{2n-2} \prod_{1 \le i < j \le n} \left( z_i - z_j \right)^2 \]
	where $z_1, \dots, z_n$ are the roots of $f$, and $c$ is the leading coefficient of $f$.

	Suppose $K$ is monogenic with $\OO_K = \ZZ[\theta]$.
	Let $f$ denote the minimal polynomial of $\theta$ (hence monic).
	Show that \[ \Delta_K = \Delta(f). \]
	\label{prob:root_discriminant}
	\begin{hint}
		Vandermonde matrices.
	\end{hint}
\end{sproblem}


\begin{problem}
	Show that if $K \neq \QQ$ is a number field then $\left\lvert \Delta_K \right\rvert > 1$.
	\begin{hint}
		$M_K \ge 1$ must hold. Bash.
	\end{hint}
\end{problem}

\begin{problem}
	[Brill's theorem]
	For a number field $K$ with signature $(r_1, r_2)$, show that
	$\Delta_K > 0$ if and only if $r_2$ is even.
\end{problem}

\begin{problem}
	[Stickelberger theorem]
	\kurumi
	Let $K$ be a number field. Prove that \[ \Delta_K \equiv 0 \text{ or } 1 \pmod 4. \]
	% P N
\end{problem}