Making sense of capitulation: reciprocal primes
Tom 172 / 2016
Streszczenie
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 $\ell$.
