Talk:Galois extension

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 Field: Algebra

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[edit]

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)