Talk:Galois extension

Artin?
Actually, I'm not totally sure about my recent edit, in particular I'm not completely sure which part of the result is due to Artin. Dmharvey 19:00, 15 April 2006 (UTC)

Highly ambiguous wording
The article reads:

''"An important theorem of Emil Artin states that a finite extension E/F is Galois if and only if any one of the following conditions holds:
 * E/F is a normal extension and a separable extension.
 * E is the 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."''

One entirely reasonable interpretation of this is of the form:

"X" is equivalent to "A or B or C".

(For example: "n belongs to the set {1,2,3}" is equivalent to "n=1 or n=2 or n=3".)

The correct statement, however, should say unambiguously:

''"Given a finite extension E/F, the following are equivalent definitions of what it means to say E/F is Galois:

1. E/F is a normal extension and a separable extension.

2. E is the splitting field of a separable polynomial with coefficients in F.

3. [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."''Daqu 21:51, 3 December 2007 (UTC)
 * I've made it unambiguous. Wording is a bit clumsy though. Algebraist 17:48, 17 February 2008 (UTC)