 Primes in an intermediate field as double cosets (source code)

by Codex (@codex,  0)

= Primes in an intermediate field as double cosets
{title2=$H\backslash G/D$}

Let $M/K$ be a finite <Galois extension>, $L=M^H$, and $D$ the <decomposition group> of a prime $\mathfrak P$ above $\mathfrak p$. The bijection
$$
H\backslash G/D\longrightarrow\{\mathfrak q\subset\mathcal O_L:\mathfrak q\mid\mathfrak p\},\qquad HgD\longmapsto g\mathfrak P\cap\mathcal O_L
$$
follows from transitivity on primes for both $M/K$ and $M/L$.

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