Editing Field (mathematics) (section)

== Galois theory ==
{{Main|Galois theory}}

Galois theory studies [[algebraic extension]]s of a field by studying the [[Symmetry group#Symmetry groups in general|symmetry]] in the arithmetic operations of addition and multiplication. An important notion in this area is that of [[finite extension|finite]] [[Galois extension]]s {{math|''F'' / ''E''}}, which are, by definition, those that are [[separable extension|separable]] and [[normal extension|normal]]. The [[primitive element theorem]] shows that finite separable extensions are necessarily [[simple extension|simple]], i.e., of the form
: {{math|1=''F'' = ''E''[''X''] / {{itco|''f''}}(''X'')}},
where {{math|''f''}} is an irreducible polynomial (as above).<ref>{{harvp|Lang|2002|loc=Theorem V.4.6}}</ref> For such an extension, being normal and separable means that all zeros of {{math|''f''}} are contained in {{math|''F''}} and that {{math|''f''}} has only simple zeros. The latter condition is always satisfied if {{math|''E''}} has characteristic {{math|0}}.

For a finite Galois extension, the [[Galois group]] {{math|Gal(''F''/''E'')}} is the group of [[field automorphism]]s of {{math|''F''}} that are trivial on {{math|''E''}} (i.e., the [[bijection]]s {{math|''σ'' : ''F'' → ''F''}} that preserve addition and multiplication and that send elements of {{math|''E''}} to themselves). The importance of this group stems from the [[fundamental theorem of Galois theory]], which constructs an explicit [[one-to-one correspondence]] between the set of [[subgroup]]s of {{math|Gal(''F''/''E'')}} and the set of intermediate extensions of the extension {{math|''F''/''E''}}.<ref>{{harvp|Lang|2002|loc=§VI.1}}</ref> By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not [[solvable group|solvable]] (cannot be built from [[abelian group]]s), then the zeros of {{math|''f''}} ''cannot'' be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving <math>\sqrt[n]{~}</math>. For example, the [[symmetric group]]s {{math|S<sub>''n''</sub>}} is not solvable for {{math|''n'' ≥ 5}}. Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the [[Abel–Ruffini theorem]]:
: {{math|1=''f''(''X'') = ''X''<sup>5</sup> − 4''X'' + 2}} (and {{math|1=''E'' = '''Q'''}}),<ref>{{harvp|Lang|2002|loc=Example VI.2.6}}</ref>
: {{math|1=''f''(''X'') = {{itco|''X''}}<sup>''n''</sup> + ''a''<sub>''n''−1</sub>{{itco|''X''}}<sup>''n''−1</sup> + ⋯ + ''a''<sub>0</sub>}} (where {{math|''f''}} is regarded as a polynomial in {{math|''E''(''a''<sub>0</sub>, ..., ''a''<sub>''n''−1</sub>)}}, for some indeterminates {{math|''a''<sub>''i''</sub>}}, {{math|''E''}} is any field, and {{math|''n'' ≥ 5}}).

The [[tensor product of fields]] is not usually a field. For example, a finite extension {{math|''F'' / ''E''}} of degree {{math|''n''}} is a Galois extension if and only if there is an isomorphism of {{math|''F''}}-algebras
: {{math|''F'' ⊗<sub>''E''</sub> ''F'' ≅ ''F''<sup>''n''</sup>}}.
This fact is the beginning of [[Grothendieck's Galois theory]], a far-reaching extension of Galois theory applicable to algebro-geometric objects.<ref>{{harvp|Borceux|Janelidze|2001}}. See also [[Étale fundamental group]].</ref>