Editing Galois theory (section)

==Modern approach by field theory==
{{Unreferenced section|date=June 2023}}

In the modern approach, one starts with a [[field extension]] {{math|''L''/''K''}} (read "{{math|''L''}} over {{math|''K''}}"), and examines the group of [[automorphism]]s of {{math|''L''}} that fix {{math|''K''}}. See the article on [[Galois group]]s for further explanation and examples.

The connection between the two approaches is as follows. The coefficients of the polynomial in question should be chosen from the base field {{math|''K''}}. The top field {{math|''L''}} should be the field obtained by adjoining the roots of the polynomial in question to the base field {{math|''K''}}. Any permutation of the roots which respects algebraic equations as described above gives rise to an automorphism of {{math|''L''/''K''}}, and vice versa.

In the first example above, we were studying the extension {{math|'''Q'''({{sqrt|3}})/'''Q'''}}, where {{math|'''Q'''}} is the field of [[rational number]]s, and {{math|'''Q'''({{sqrt|3}})}} is the field obtained from {{math|'''Q'''}} by adjoining {{math|{{sqrt|3}}}}. In the second example, we were studying the extension {{math|'''Q'''(''A'',''B'',''C'',''D'')/'''Q'''}}.

There are several advantages to the modern approach over the permutation group approach.
* It permits a far simpler statement of the [[fundamental theorem of Galois theory]].
* The use of base fields other than {{math|'''Q'''}} is crucial in many areas of mathematics. For example, in [[algebraic number theory]], one often does Galois theory using [[number field]]s, [[finite field]]s or [[local field]]s as the base field.
* It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses the [[absolute Galois group]] of {{math|'''Q'''}}, defined to be the Galois group of {{math|''K''/'''Q'''}} where {{math|''K''}} is an [[algebraic closure]] of {{math|'''Q'''}}.
* It allows for consideration of [[inseparable extension]]s. This issue does not arise in the classical framework, since it was always implicitly assumed that arithmetic took place in [[characteristic (algebra)|characteristic]] zero, but nonzero characteristic arises frequently in number theory and in [[algebraic geometry]].
* It removes the rather artificial reliance on chasing roots of polynomials. That is, different polynomials may yield the same extension fields, and the modern approach recognizes the connection between these polynomials.