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{{r|Field extension}}

{{r|Field extension}}

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

A field extension of the rational numbers of finite degree; a principal object of study in al

{{r|Field extension}}

{{r|Field extension}}

...s the [[ramification#In algebraic number theory|ramification]] data of the field extension ''L''/''K''. A prime ideal ''p'' of ''K'' ramifies in ''L'' if and only if

{{r|Field extension}}

...bgroup <math>Aut_L(K)</math> of the full automorphism group of ''K''. A [[field extension]] <math>K/L</math> of finite index ''d'' is ''[[normal extension|normal]]''

{{r|Field extension}}

..., a '''splitting field''' for a polynomial ''f'' over a field ''F'' is a [[field extension]] ''E''/''F'' with the properties that ''f'' splits completely over ''E'',

...''n''-th root of unity, then the ''n''-th cyclotomic field ''F'' is the [[field extension]] <math>\mathbf{Q}(\zeta)</math>.

...quadratic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] of its [[prime field]] of degree two.

{{r|Field extension}}

{{r|Field extension}}

...The degree of the minimal polynomial of α is equal to the degree of the [[field extension]] '''Q'''(α)/'''Q'''. ...tisfies. The polynonmial ring ''F''[α] is then a field, and is the simple field extension ''F''(α). This field is a finite dimension al vector space over ''F'', on

Key concepts are [[Field extension|field extensions]] and [[Group theory|groups]], which should be thoroughly ...a subfield and also all the roots of ''f''. This field is known as the [[field extension|extension]] of ''K'' by the roots of ''f'', or the ''[[splitting field]]''

{{r|Field extension}}

{{r|Field extension}}

*[[Field extension]]