Making sense of capitulation: reciprocal primes

• Autorzy: David Folk
• Języki publikacji: EN

Abstrakty

EN

Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, 𝔭 a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and $a_{𝔭}$ any generator of the principal ideal $𝔭^{ℓ}$. We will call a prime ideal 𝔮 of K 'reciprocal to 𝔭' if its Frobenius element generates $Gal(K(\sqrt[ℓ]{a_{𝔭}})/K)$ for every choice of $a_{𝔭}$. We then show that 𝔭 becomes principal in L if and only if every reciprocal prime 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 ℓ.

Kategorie tematyczne

• 11R37: Class field theory
• 11R29: Class numbers, class groups, discriminants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 172
• Numer: 4
• Strony: 325-332

Daty

wydano

2016

Twórcy

autor

David Folk

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

Identyfikatory

DOI

10.4064/aa8264-1-2016