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• # Artykuł - szczegóły

## Acta Arithmetica

2016 | 172 | 4 | 325-332

## Making sense of capitulation: reciprocal primes

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.

325-332

wydano
2016

### Twórcy

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