Search results

A formal product of places of an algebraic number field, used to encode ramification data for abelian extensions of a number

Auto-populated based on [[Special:WhatLinksHere/Modulus (algebraic number theory)]]. Needs checking by a human. {{r|Algebraic number field}}

{{r|Algebraic number field}} {{r|Algebraic number}}

{{r|Algebraic number}}

{{r|Algebraic number field}} {{r|Modulus (algebraic number theory)}}

{{r|Algebraic number field}} {{r|Algebraic number}}

<noinclude>{{Subpages}}</noinclude>The branch of algebraic number theory which studies the abelian extensions of a number field, or more gene

An invariant attached to an extension of algebraic number fields which encodes ramification data.

An algebraic number field for which the ring of integers is a polynomial ring.

...ted in algebraic number theory, performing sophisticated computations in [[algebraic number field]]s, in [[Global field|global]] [[function field]]s, and in [[local fi ...k | author=J. Graf von Schmettow | title=KANT — a tool for computations in algebraic number fields | booktitle=Computational number theory | publisher=de Gruyter | yea

An element of an algebraic number field which has a denominator confined to primes in some fixed set.

{{r|Algebraic number field}} {{r|Discriminant of an algebraic number field}}

{{r|Algebraic number}}

...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub *{{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=[[Springer-Verlag]] | isbn=0-387-94225-4 | year=1986 }}

| title = Algebraic Number Theory

{{r|Algebraic number field}} {{r|Algebraic number}}

Auto-populated based on [[Special:WhatLinksHere/Discriminant of an algebraic number field]]. Needs checking by a human. {{r|Algebraic number field}}

In [[algebraic number theory]], the '''genus field''' ''G'' of a [[number field]] ''K'' is the [[ * {{cite book | last=Ishida | first=Makoto | title=The genus fields of algebraic number fields | series=Lecture Notes in Mathematics | publisher=[[Springer Verlag]

...lgebraic, but the converse need not hold. For example, the field of all [[algebraic number]]s over '''Q''' is an algebraic extension but not of finite degree.

* {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }}

{{r|Modulus (algebraic number theory)}}

...n of the rational numbers of finite degree; a principal object of study in algebraic number theory.

...field''' is an invariant attached to an [[field extension|extension]] of [[algebraic number field]]s which describes the geometric structure of the [[ring of integers]

Roots of unity are clearly [[algebraic number]]s, and indeed [[algebraic integer]]s. It is often convenient to identify

An invariant attached to an extension of algebraic number fields which describes the geometric structure of the ring of integers and

{{r|Algebraic number field}} {{r|Algebraic number}}

In [[mathematics]], a '''monogenic field''' is an [[algebraic number field]] for which there exists an element In a monogenic field ''K'', the [[Discriminant of an algebraic number field|field discriminant]] of ''K'' is equal to the [[discriminant of a pol

...c embedding of the generators of the unit group of the maximal order of an algebraic number field.

...ink=J. W. S. Cassels | coauthors=[[Albrecht Fröhlich|A. Fröhlich]] | title=Algebraic Number Theory | publisher=[[Academic Press]] | year=1967 | isbn=012268950X }} * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=[[Springer-Verlag]] | isbn=0-387-94225-4 | year=1986 }}

...ink=J. W. S. Cassels | coauthors=[[Albrecht Fröhlich|A. Fröhlich]] | title=Algebraic Number Theory | publisher=[[Academic Press]] | year=1967 | isbn=012268950X }} * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=[[Springer-Verlag]] | isbn=0-387-94225-4 | year=1986 }}

{{r|Algebraic number}}

{{r|Algebraic number field}} {{r|Algebraic number}}

In [[mathematics]], and more specifically&mdash;in [[number theory]], an '''algebraic number''' is a [[complex number]] that is a root of a [[polynomial]] with [[ration ...s that ensued forms the foundation of modern [[algebraic number theory]]. Algebraic number theory is now an immense field, and one of current research, but so far has

{{r|Algebraic number}}

{{r|Algebraic number field}} {{r|Algebraic number}}

{{r|Algebraic number}}

{{r|Discriminant of an algebraic number field}}

{{r|Algebraic number}}

...e of integral closure is the [[ring of integers]] or maximal order in an [[algebraic number field]] ''K'', which may be defined as the integral closure of '''Z''' in ' * {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }}

...], the '''different ideal''' is an invariant attached to an extension of [[algebraic number field]]s. ...tive norm]] of the relative different is equal to the [[Discriminant of an algebraic number field|relative discriminant]] Δ_''L''/''K''. In a tower of fiel

...of a [[square matrix]], an [[endomorphism]] of a [[vector space]] or an [[algebraic number]]. ==Minimal polynomial of an algebraic number==

...'''cycle''') is a formal product of [[Place (mathematics)|place]]s of an [[algebraic number field]]. It is used to encode [[ramification]] data for [[abelian extensio Let ''K'' be an algebraic number field with ring of integers ''R''. A ''modulus'' is a formal product

{{r|Algebraic number field}}

{{r|Algebraic number field}}

{{r|Algebraic number field}}

{{r|Algebraic number field}}

In [[mathematics]], to each [[algebraic number field]] ''k'', there is associated an important function called the '''Dede If ''k'' is an algebraic number field, the Dedekind zeta function of the field is a [[meromorphic function]

...ory]], an '''algebraic number field''' is a principal object of study in [[algebraic number theory]]. The algebraic and arithmetic structure of a number field has app ...field]] '''Q''' of [[rational number]]s. The elements of ''K'' are thus [[algebraic number]]s. Let ''n'' = [''K'':'''Q'''] be the degree of the extension.

{{r|Algebraic number field}}

{{r|Algebraic number field}}