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...ion|extensions]] of an abelian group. Every finite extension of a [[finite field]] is a cyclic extension. ...|function fields]] of [[algebraic curve]]s over finite fields, and [[local field]]s. ...

'''Trivial extension''' may refer to the following types of extensions: *A [[trivial field extension]] ...

...{{harvs|txt|authorlink=Ernst Witt|last=Witt|year=1936}} generalized it to extensions of prime power degree ''p''^''n''. If ''K'' is a [[field (mathematics)|field]] of characteristic ''p'', a [[prime number]], any [[polynomial]] of the fo ...

...particular kind, which is a [[Galois extension]] of the [[rational number field]] {{mvar|ℚ}} with [[Galois group]] isomorphic to the [[Klein four-gro According to [[Galois theory]], there must be three [[quadratic field]]s contained in {{mvar|K}}, since the Galois group has three [[subgroup]]s ...

These topics are basic to the field, either as prototypical examples, or as basic objects of study. *[[Algebraic number field]] ...

...matics]], a [[polynomial]] ''P''(''X'') over a given [[field (mathematics)|field]] ''K'' is '''separable''' if its [[root of a polynomial|roots]] are [[dist ...neral, ''P''(''X'') is separable if and only if it is square-free over any field that contains ''K'', ...

...s group remains.'' From {{SpringerEOM|id=Class_field_theory | title= Class field theory | oldid=17596 | first=L.V. | last=Kuz'min}}.</ref> ...extension]]; a common way to explain the objective of a non-abelian class field theory is that it should provide a more explicit way to express such patter ...

...)|adjunction]] of a single element, called a ''primitive element''. Simple extensions are well understood and can be completely classified. ...em]] provides a characterization of the [[finite extension|finite]] simple extensions. ...

{{short description|Correspondence between finite abelian extensions and generalized ideal class groups}} ...ld theory]], the '''Takagi existence theorem''' states that for any number field ''K'' there is a one-to-one inclusion reversing correspondence between the ...

...X)</math> over a given [[field (mathematics)|field]] <math>K</math> is a [[field extension]] of <math>K</math> generated by a [[root of a function|root]] <m ...math>P(X)=X^3-2</math> then <math>\mathbb Q[\sqrt[3]2]</math> is a rupture field for <math>P(X)</math>. ...

...te<ref>Roman, p. 42.</ref> of <math>E_1</math> and <math>E_2</math> is the field defined as the intersection of all subfields of ''K'' containing both <math It also can be defined using [[field of fractions]] ...

...'e'' is the largest. Now any common model with an embedding from these two extensions must be at least of size five so that there are two elements on either sid ...mmon field. However, two arbitrary fields cannot be embedded into a common field when the [[Characteristic (algebra)|characteristic]] of the fields differ. ...

* [[Field extension]], in Galois theory * [[Extensions (Ahmad Jamal album)|''Extensions'' (Ahmad Jamal album)]], 1965 ...

...' in ''k''. Purely inseparable extensions are sometimes called '''radicial extensions''', which should not be confused with the similar-sounding but more general ==Purely inseparable extensions== ...

{{Short description|Extension of a mathematical field with polynomial roots}} .../ref><ref>Malik, Mordeson, Sen (1997), Definition 21.1.23, p. 453.</ref> A field extension that is not algebraic, is said to be [[Transcendental extension|t ...

{{short description|Class of extensions of the real numbers}} ...theory]], and the study of [[Banach algebra]]s. The [[Field (mathematics)|field]] of superreals is itself a subfield of the [[surreal number]]s. ...

{{Short description|Type of algebraic field extension}} ...sion|finite extensions]], a normal extension is identical to a [[splitting field]]. ...

* Wi-Fi Multimedia or [[Wireless Multimedia Extensions]] (WME), a QoS implementation * [[World Magnetic Model]], a digital model of the Earth's geomagnetic field ...

{{Short description|Algebraic field extension}} ...'K'' is an [[algebraic extension]] of ''K'' that is [[algebraically closed field|algebraically closed]]. It is one of many [[closure (mathematics)|closures ...

In [[mathematics]], in the field of [[group theory]], a [[subgroup]] of a [[group (mathematics)|group]] is s ...a conjugacy-closed subgroup of the general linear group over the extension field. This result is typically referred to as a '''stability theorem'''. ...