Class Field Theory

Neukirch, Jürgen

doi:10.1007/978-3-642-82465-4

Cited by 140 publications

(71 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…By Proposition 1.10 in [8] there exists a prime p in the class of x splitting completely in E k,µ,τ l c /k. By Theorem IV.8.4 in [19], p ∈ H m E k,µ,τ l c /k , where m is a cycle of declaration of E k,µ,τ l c /k. Then, by Proposition II.3.3 in [19],…”

mentioning

confidence: 92%

Steinitz classes of tamely ramified nonabelian extensions of odd prime power order

Cobbe

2011

Acta Arith.

View full text Add to dashboard Cite

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O k of k, which, together with the degree [K : k] of the extension determines the O k -module structure of O K . We call R t (k, G) the classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group.In this paper we will develop some of the ideas contained in [8] to study some l-groups, where l is an odd prime number. In particular, together with [1] we will complete the study of realizable Steinitz classes for groups of order l 3 . We will also give an alternative proof of the results of [1], based on class field theory. R t (k, G) = {x ∈ Cl(k) : ∃K/k tame, Gal(K/k) ∼ = G, st(K/k) = x}.

show abstract

mentioning

confidence: 92%

Steinitz classes of tamely ramified nonabelian extensions of odd prime power order

Cobbe

2011

Acta Arith.

View full text Add to dashboard Cite

show abstract

“…To fix notation, we give a quick review of basic class field theory [25]. Let K be a number field, and let J K be the free group generated by the finite primes of K. There is a natural map ι : K × → J K .…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…Let K be a number field, and let J K be the free group generated by the finite primes of K. There is a natural map ι : K × → J K . A modulus, called a cycle in [25], is a finite formal product of primes of K with non-negative exponents p p n p . If m = p p n p is a modulus, and x ∈ K, we write x ≡ 1 mod m to mean:…”

Section: Proof Of Propositionmentioning

confidence: 99%

See 1 more Smart Citation

Distribution of orders in number fields

Kaplan

Marcinek

Takloo-Bighash

2015

Mathematical Sciences

View full text Add to dashboard Cite

In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techniques to analyze the zeta function. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field. We also obtain non-trivial bounds for higher degree number fields.

show abstract

“…Needless to say, there are many other important approaches to local class field theory, see e.g. [3], [6], [8], [12], [15], and [16].…”

Section: The Hasse-arf Theoremmentioning

confidence: 99%