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*[[Field extension]]

.../math> of fields such that for all <math> i > 0 , M_i </math> is a normal field extension [[Galois theory glossary|(glossary)]] of <math> M_{i-1} </math>. ...m the subgroup structure of <math>S_2</math> that there is no intermediate field extension containing Q and also roots of the polynomial.

In Δ is not a square in ''F'' then the [[field extension]] <math>F(\sqrt\Delta)</math> is [[quadratic field|quadratic]] over ''F'':

Let us take the case that ''G'' is the [[Galois group]] of a [[field extension]] ''L''/''K''. A factor system in H²(''G'',''L''^*) g

An ''algebraic number field'' ''K'' is a finite degree [[field extension]] of the [[field (mathematics)|field]] '''Q''' of [[rational number]]s. Th

...e one. Since <math>\scriptstyle\mathbb{C} = \mathbb{R}[i]</math>, any such field extension also extends <math>\scriptstyle\mathbb{R}</math>. Now, any <math>\scriptsty

* [[Algebraically independent set]]s in a [[field extension]];

Let ''K'' be an [[algebraic number field]], a finite [[field extension|extension]] of '''Q''', and ''E'' an elliptic curve defined over ''K''. Th

...though in this case what is left invariant is the equation(s) defining the field extension. I'm less convinced that this is a useful motivating example, but it is ''t

...analysis, we could next show that <math>\mathbb{C}</math> has no finite [[field extension|extension]] and must therefore be [[algebraic closure|algebraically closed]

...analysis, we could next show that <math>\mathbb{C}</math> has no finite [[field extension|extension]] and must therefore be [[algebraic closure|algebraically closed]

*12Fxx [[Field extension]]s

...tain (real) roots for those polynomials. Now, the Galois group of a normal field extension (roughly, one that arises through adjunction of all roots of a set of polyn ...in this analysis, we discover that <math>\mathbb{C}</math> has no finite [[field extension|extension]] and must therefore be [[algebraic closure|algebraically closed]