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Galois extension

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In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.

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A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.

Characterization of Galois extensions

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.
  • |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.
  • |Aut(E/F)| ≥ [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 Aut(E).
  • F is the fixed field of Aut(E/F).
  • There is a one-to-one correspondence between subfields of E/F and subgroups of Aut(E/F).

    • Examples

    There are two basic ways to construct examples of Galois extensions.

  • Take any field E, any subgroup of 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 cube 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 x2 − 2; the second has normal closure that includes the complex cube 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 x3 − 2 has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.

    An algebraic closure K ¯ of an arbitrary field K is Galois over K if and only if K is a perfect field.

    References

    Galois extension Wikipedia
    (Text) CC BY-SA