Crystal system
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. One says that such extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. [1]
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, then E/F is a Galois extension, where F is the fixed field of G.
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.
- [E:F] = |Aut(E/F)|; that is, the degree of the field extension is equal to the order of the automorphism group of E/F.
Examples
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.
An algebraic closure of an arbitrary field is Galois over if and only if is a perfect field.
See also
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Later republished in English by Springer under the title "Algebra".) - Template:Cite web
References
- ↑ See the article Galois group for definitions of some of these terms and some examples.