Galois extension

An important theorem of Emil Artin states that for a finite extension E/F, each of the following statements is equivalent to the statement that E/F is Galois: • E/F is a normal extension and a separable extension. • E is a splitting field of a separable polynomial with coefficients in F. • |\!\operatorname{Aut}(E/F)| = [E:F], that is, the number of automorphisms equals the degree of the extension. Other equivalent statements are: • Every irreducible polynomial in F[x] with at least one root in E splits over E and is separable. • |\!\operatorname{Aut}(E/F)| \geq [E:F], that is, the number of automorphisms is at least the degree of the extension. • F is the fixed field of a subgroup of \operatorname{Aut}(E). • F is the fixed field of \operatorname{Aut}(E/F). • There is a one-to-one correspondence between subfields of E/F and subgroups of \operatorname{Aut}(E/F). An infinite field extension E/F is Galois if and only if E is the union of finite Galois subextensions E_i/F indexed by an (infinite) index set I, i.e. E=\bigcup_{i\in I}E_i and the Galois group is an inverse limit \operatorname{Aut}(E/F)=\varprojlim_{i\in I}{\operatorname{Aut}(E_i/F)} where the inverse system is ordered by field inclusion E_i\subset E_j. ==Examples==

There are two basic ways to construct examples of Galois extensions. • Take any field E, any finite subgroup of \operatorname{Aut}(E), and let F be the fixed field. • Take any field F, any separable polynomial in F[x], and let E be its splitting field. Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cubic root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first of them is the splitting field of x^2 -2; the second has normal closure that includes the complex cubic roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and x^3 -2 has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory. An algebraic closure \bar K of an arbitrary field K is Galois over K if and only if K is a perfect field. == Notes ==