10.1007/978-3-662-03983-0

Neukirch, J. (1999). 10.1007/978-3-662-03983-0. Algebraic Number Theory.

Type

Book

Intellectual base

Review or monograph

Year

1999

DOI

10.1007/978-3-662-03983-0

ISBN

9783642084737

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-3-662-03983-0

Cited by

8 citations across 8 artifacts.

Each citation links to the sentence it supports in the citing article.

Domain-specific

• Algebraic number field
  • An algebraic number field is a field extension K/Q of finite degree, equivalently a field generated over Q by finitely many algebraic numbers and, by the primitive element theorem in characteristic zero, by one algebraic number.
• Biquadratic field
  • … from A canonical instance directly demonstrates that the two radicands represent independent nontrivial rational square classes and the resulting extension has degree four and V4 Galois group. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, or implementation..
• Dedekind zeta function
  • The Dedekind zeta function is ζ_K(s)=Σ_{I≠0}N(I)^{-s}, initially for Re(s)>1, equivalently an Euler product over prime ideals.
• Hecke Character
  • Its values on uniformizers generate an Euler product and Hecke L-function, extending Dirichlet L-functions from \(\mathbb Q\) to general number fields.
• Local field
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Norm group
  • … field K, finite abelian extension L and degree, multiplicative groups, field norm formula, image subgroup, openness and index, reciprocity map and extension-subgroup correspondence are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Profinite Integer
  • The absolute Galois group of a finite field is canonically isomorphic to \(\widehat{\mathbb Z}\) as a profinite group, with Frobenius supplying a topological generator in the procyclic sense.

    This sourceReference for local fields, Galois groups, and arithmetic completions.

• Ring
  • Abstract algebra — the home turf: the ring is a fundamental object, with ideal theory, quotient rings, prime/maximal ideals, and the first isomorphism theorem proved at this tier. Algebraic number theory — rings of algebraic integers, ideals, factorization, and class groups carry the entire edifice of modern number theory inside ring theory

    This sourceRings of algebraic integers, prime-ideal factorization in Dedekind domains, and the ideal class group as the foundations of algebraic number theory (Ch. I).

    Supported in partVerified against the source

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