Talk:Galois extension

Latest comment: 16 years ago by Algebraist in topic Highly ambiguous wording

Artin?

edit

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)Reply

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)Reply

I've made it unambiguous. Wording is a bit clumsy though. Algebraist 17:48, 17 February 2008 (UTC)Reply