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...ional numbers '''Q''', such that the [[discriminant of an algebraic number field|discriminant]] of ''K''/'''Q''' is at most ''N''. The theorem is named aft ...'Q''' has no [[Splitting of prime ideals in Galois extensions|unramified]] extensions. ...

{{Short description|In mathematics, a sequence of field extensions}} In [[mathematics]], a '''tower of fields''' is a sequence of [[field extension]]s ...

...sential because it allows one to construct analogues of [[algebraic number field]]s in the p-adic context. === Finite field extensions === ...

In mathematics, '''class field theory''' is the study of abelian extensions of local and global fields. ...Abel]] uses special values of the lemniscate function to construct abelian extensions of <math>\mathbb{Q}(i)</math>. ...

...bhyankar's inequality''' is an inequality involving extensions of [[valued field]]s in [[algebra]], introduced by {{harvs|txt|authorlink=Shreeram Shankar Ab ...] of ''K''/''k'' is at least the [[transcendence degree]] of the [[residue field]] extension plus the rank of the [[quotient]] of the [[valuation group]]s; ...

...s [[isomorphism|isomorphic]] to a finite product of finite separable field extensions. An étale algebra is a special sort of commutative [[separable algebra]]. Let {{mvar|K}} be a [[field (mathematics)|field]]. Let {{mvar|L}} be a [[commutative]] [[unital algebra|unital]] [[associa ...

{{Short description|Mathematical field obtained by adjunction of nth roots}} ...a '''radical extension''' of a [[field (mathematics)|field]] ''K'' is an [[field extension|extension]] of ''K'' that is obtained by adjoining a sequence of ...

{{short description|Iwasawa's μ-invariant is 0 for cyclotomic extensions of abelian algebraic number fields}} | field = [[Algebraic number theory]] ...

...cation''' of a [[prime ideal]] can often be reduced to the case of [[local field]]s where a more detailed analysis can be carried out with the aid of tools In this article, a local field is non-archimedean and has finite [[residue field]]. ...

...olume=LI |issue=7a |pages=1795–1815}}</ref> is a [[non-Archimedean ordered field]]; i.e., a system of numbers containing infinite and [[infinitesimal]] quan The [[real number]]s are embedded in this field as series in which all of the coefficients vanish except <math>a_0</math>. ...

If ''A''&nbsp;=&nbsp;''k'' is a [[field (mathematics)|field]], the Bass–Quillen conjecture asserts that any projective module over <mat ...in the case that ''A'' is a smooth [[algebra over a field|algebra]] over a field ''k''. Further known cases are reviewed in {{Harvtxt|Lam|2006}}. ...

...to K</math>. A difference algebra that is a field is called a ''difference field extension''. ...polynomial ring <math>K\{y\}=K\{y_1,\ldots,y_n\}</math> over a difference field <math>K</math> in the (difference) variables <math>y_1,\ldots,y_n</math> is ...

In [[field theory (mathematics)|field theory]], a branch of algebra, a [[field extension]] <math>L/k</math> is said to be '''regular''' if ''k'' is [[alge * Any extension of an algebraically closed field is regular.<ref name=FJ39/><ref name=C426>Cohn (2003) p.426</ref> ...

...symbol takes values in the ''m'' roots of 1. When ''m'' = 2 and the global field is the rationals this is more or less the same as the Jacobi symbol. ...frac{a,b}{p}\right)_m</math> is defined for ''a'' and ''b'' in some global field ''K'', for ''p'' a finite or infinite place of ''K'', and is equal to the l ...

...|Relates the topology of a complete non-archimedean field to its algebraic extensions}} ...ogical space|topology]] of a [[complete field|complete]] [[non-archimedean field]] to its [[algebraic extension]]s. ...

...thematics]], a '''quadratically closed field''' is a [[field (mathematics)|field]] of [[Characteristic (algebra)|characteristic]] not equal to 2 in which ev ...ber]]s is quadratically closed; more generally, any [[algebraically closed field]] is quadratically closed. ...

...nally stated that, assuming that <math>F</math> is a [[totally real number field]] and that <math>F_\infty/F</math> is the cyclotomic <math>\mathbb{Z}_p</ma ...ell</math> is a fixed prime, with consideration of subfields of cyclotomic extensions of <math>k</math>, one can define a tower of number fields ...

In mathematics, a [[Field (mathematics)|field]] ''K'' with an [[Absolute value#Fields|absolute value]] is called '''spher The definition can be adapted also to a field ''K'' with a [[Valuation (algebra)|valuation]] ''v'' taking values in an ar ...

...thematics)|field]] describes the structure of [[quadratic form]]s over the field. ...ic space]] over ''F'', or ∞ if this does not exist. Since [[formally real field]]s have anisotropic quadratic forms (sums of squares) in every dimension, t ...

...nd <math>Z_p</math> extensions''.<ref>''Class numbers and <math>Z_p</math> extensions'', Mathematische Annalen, vol. 214, 1975, p. 177</ref> He then became an as Washington wrote a standard work on [[cyclotomic field]]s. He also worked on [[p-adic]] [[L-function]]s. He wrote a treatise with ...