root-discriminantRootDiscriminant2005-02-24 16:34:572005-02-24 16:37:42Definitiondiscriminantdiscriminantroot discriminant% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\rd}{\operatorname{rd}}\begin{defn}
Let $K$ be a number field, let $d_K$ be its discriminant and let $n=[K:\Rats]$ be the degree over $\Rats$. The quantity:
$$|\sqrt[n]{d_K}|$$
is called the {\bf root-discriminant} of $K$ and it is usually denoted by $\operatorname{rd}_K$.
\end{defn}
The following lemma is one of the motivations for the previous definition:
\begin{lemma}
Let $E/F$ be an extension of number fields which is unramified at all finite primes. Then $\rd_E=\rd_F$. In particular, the Hilbert class field of a number field has the same root-discriminant as the number field.
\end{lemma}
\begin{proof}
Notice that the relative discriminant ideal (or different) for $E/F$ is the ring of integers in $F$. Therefore we have:
$$|d_E|=|d_F|^{[E:F]}$$
The results follows by taking $[E:\Rats]$-th roots on both sides of the previous equation.
\end{proof}