Making sense of capitulation: reciprocal primes

David Folk

Acta Arithmetica (2016)

  • Volume: 172, Issue: 4, page 325-332
  • ISSN: 0065-1036

Abstract

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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 a any generator of the principal ideal ℓ . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates G a l ( K ( a ℓ ) / K ) for every choice of a . 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.

How to cite

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David Folk. "Making sense of capitulation: reciprocal primes." Acta Arithmetica 172.4 (2016): 325-332. <http://eudml.org/doc/278967>.

@article{DavidFolk2016,
abstract = {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 $a_\{\}$ any generator of the principal ideal $^\{ℓ\}$. We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $Gal(K(\@root ℓ \of \{a_\{\}\})/K)$ for every choice of $a_\{\}$. 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.},
author = {David Folk},
journal = {Acta Arithmetica},
keywords = {capitulation; principalization},
language = {eng},
number = {4},
pages = {325-332},
title = {Making sense of capitulation: reciprocal primes},
url = {http://eudml.org/doc/278967},
volume = {172},
year = {2016},
}

TY - JOUR
AU - David Folk
TI - Making sense of capitulation: reciprocal primes
JO - Acta Arithmetica
PY - 2016
VL - 172
IS - 4
SP - 325
EP - 332
AB - 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 $a_{}$ any generator of the principal ideal $^{ℓ}$. We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $Gal(K(\@root ℓ \of {a_{}})/K)$ for every choice of $a_{}$. 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.
LA - eng
KW - capitulation; principalization
UR - http://eudml.org/doc/278967
ER -

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