Making sense of capitulation: reciprocal primes

David Folk¹

¹ Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197, U.S.A.

Acta Arithmetica, Tome 172 (2016) no. 4, pp. 325-332.

Résumé

Let $\ell$ be a rational prime, $K$ be a number field that contains a primitive $\ell$th root of unity, $L$ an abelian extension of $K$ whose degree over $K$, $[L:K]$, is divisible by $\ell$, $\mathfrak{p}$ a prime ideal of $K$ whose ideal class has order $\ell$ in the ideal class group of $K$, and $a_{\mathfrak{p}}$ any generator of the principal ideal $\mathfrak{p}^{\ell}$. We will call a prime ideal $\mathfrak{q}$ of $K$ ‘reciprocal to $\mathfrak{p}$’ if its Frobenius element generates Gal$(K(\sqrt[\ell]{a_{\mathfrak{p}}})/K)$ for every choice of $a_{\mathfrak{p}}$. We then show that $\mathfrak{p}$ becomes principal in $L$ if and only if every reciprocal prime $\mathfrak{q}$ is not a norm inside a specific ray class field, whose conductor is divisible by primes dividing the discriminant of $L/K$ and those dividing the rational prime $\ell$.

Détail

DOI : 10.4064/aa8264-1-2016

Keywords: ell rational prime number field contains primitive ellth root unity abelian extension whose degree divisible ell mathfrak prime ideal whose ideal class has order ell ideal class group mathfrak generator principal ideal mathfrak ell call prime ideal mathfrak reciprocal mathfrak its frobenius element generates gal sqrt ell mathfrak every choice mathfrak mathfrak becomes principal only every reciprocal prime mathfrak norm inside specific ray class field whose conductor divisible primes dividing discriminant those dividing rational prime ell