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, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and any generator of the principal ideal . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates for every choice of . We then show that becomes principal in L if and only if every reciprocal prime 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8264-1-2016,
author = {David Folk},
title = {Making sense of capitulation: reciprocal primes},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {325-332},
zbl = {06574965},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8264-1-2016}
}
David Folk. Making sense of capitulation: reciprocal primes. Acta Arithmetica, Tome 172 (2016) pp. 325-332. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8264-1-2016/