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{{DISPLAYTITLE:''K''-groups of a field}} ...[algebraic K-theory|algebraic ''K''-theory]], the '''algebraic ''K''-group of a field''' is important to compute. For a finite field, the complete calcul ...

...ntinuous multilinear alternating forms on the Lie algebra of smooth vector fields where the latter is given the <math>C^{\infty}</math> topology. ...irst2=D. B. |title=Cohomologies of Lie algebra of tangential vector fields of a smooth manifold |journal=Funct Anal Its Appl |volume=3 |pages=194–210 |ye ...

...ub> is a quotient of the [[polynomial ring]] '''Z'''[''X''] and the powers of ''a'' constitute a '''power integral basis'''. ...riminant]] of the [[Minimal polynomial (field theory)|minimal polynomial]] of α. ...

...(LRT) is a generalization of the [[X-ray transform]] to symmetric tensor fields <ref name="sharaf"> V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,{{isbn|90-6764-165-0}}. Chapter 2.[http://www.math.nsc.ru/~sharaf ...

In mathematics, a '''linked field''' is a [[Field (algebra)|field]] for which the [[quadra ...to Quadratic Forms over Fields | volume=67 | series=[[Graduate Studies in Mathematics]] | first=Tsit-Yuen | last=Lam | author-link=T. Y. Lam | publisher=[[Americ ...

'''Abhyankar's inequality''' is an inequality involving extensions of [[valued field]]s in [[algebra]], introduced by {{harvs|txt|authorlink=Shre ...us the rank of the [[quotient]] of the [[valuation group]]s; here the rank of an abelian group <math>A</math> is defined as <math>\dim_{\mathbb{Q}}(A \ot ...

==In mathematics== ...ory)|trees]] with seven labeled nodes.<ref>{{Cite OEIS|A000272|name=Number of trees on n labeled nodes: n^(n-2)}}</ref> ...

{{Short description|Maximal abelian extension of an algebraic number field}} ...degree [''Γ(K)'':''K''] and the '''genus group''' is the [[Galois group]] of ''Γ(K)'' over ''K''. ...

{{short description|Analogue of Stickelberger's theorem for real abelian fields}} ...o prove that some [[Tate–Shafarevich group]]s are finite, and in the proof of [[Mihăilescu's theorem]] {{harv|Schoof|2008}}. ...

...ps]] vanish up to torsion:<ref>Conjecture 51 in {{cite book|title=Handbook of K-Theory I|year=2005|publisher=Springer|pages=351–428|author=Kahn, Bruno|ed ==Finite fields== ...

{{short description|For any integer N there are only finitely many number fields with discriminant at most N}} ..., such that the [[discriminant of an algebraic number field|discriminant]] of ''K''/'''Q''' is at most ''N''. The theorem is named after [[Charles Hermi ...

{{short description|Algebraic number fields are determined by their absolute Galois groups}} In [[mathematics]], the '''Neukirch–Uchida theorem''' shows that all problems about [[algebr ...

...xtension|simple]] if and only if there are only finitely many intermediate fields between <math>K</math> and <math>L</math>. ...math>) simply the degree of <math>g</math>. Therefore, by multiplicativity of degree, <math>[M:M'] = 1</math> and hence <math>M = M'</math>. ...

==In mathematics== *the sum of four consecutive primes (<math>101+103+107+109</math>). ...

...n which generalises the idea of exponential functions on the ordered field of real numbers. ...splay="inline">K</math> onto the multiplicative group of positive elements of <math display="inline">K</math>. The ordered field <math>K\,</math> togethe ...

...e abelian extension of a number field is contained in one of its ray class fields. ...fined using positivity conditions, and uses "Strahlklasse" to mean a coset of this group. ...

...book.....C |author=Subrahmanyan Chandrasekhar |series=International Series of Monographs on Physics |publisher=Oxford: Clarendon |year=1961 |at=See discu this <math>\mathbf{F}</math> can be expressed as the sum of a toroidal field <math>\mathbf{T}</math> and poloidal vector field <math>\m ...

...very [[decreasing sequence]] of [[Ball (mathematics)|balls]] (in the sense of the metric induced by the absolute value) is nonempty:<ref>{{Cite journal | ...d abelian group: (''K'',''v'') is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection. ...

...a system of [[polynomial equations]] must have a solution in the [[Field (mathematics)|field]]. The concept is named for [[C. C. Tsen]], who introduced their st ...''F'' is a '''T'''_'''''i'''''-'''field''' if every such system, of degrees ''d''₁,&nbsp;...,&nbsp;''d''_''m'' has a commo ...

...nifold|smooth manifold]] <math>M</math>, is a generalization of the notion of a [[vector field]] on a manifold. ...>X</math> of the ''k''th [[exterior power]] <math>\wedge^k TM \to M</math> of the [[tangent bundle]], i.e. <math>X</math> assigns to each point <math>p \ ...