|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
Highly ambiguous wording
The article reads:
"The following conditions are equivalent to L / K being a normal extension:
- Let be an algebraic closure of K containing L. Every embedding σ of L in Ka which restricts to the identity on K, satisfies σ(L) = L. In other words, σ is an automorphism of L over K.
- Every irreducible polynomial in K[X] which has a root in L factors into linear factors in L[X]."
This might be saying that the conjunction of these two conditions is what's equivalent to L / K being a normal extension. Then again it might be saying that each of the two conditions is equivalent to L / K being a normal extension.
It's important to avoid such ambiguity.Daqu 21:49, 3 December 2007 (UTC)
There are three candidates for the definition of normality of a field extension L/K:
- L is the splitting field of a family of polynomials in K[X]
- Let Ka be an algebraic closure of K containing L. Every embedding σ of L in Ka which restricts to the identity on K, satisfies σ(L) = L
- Every irreducible polynomial in K[X] which has a root in L factors into linear factors in L[X]
These are equivalent for algebraic extensions but in many sources I've looked at (e.g. PlanetMath, MathReference, (Stewart 1989)) it is condition 3 that is usually given as the definition for normality, and the definition isn't restricted to algebraic extensions. This is somewhat important since the conditions don't agree for non-algebraic extensions. For example, consider the extension Q(t)/Q, where Q is the field of rational numbers and Q(t) is the field of rational functions in Q. Then no irreducible polynomial in Q[X] has a root in Q(t) and, for this reason, the extension Q(t)/Q trivially satisfies condition 3 but fails to satisy condition 1. It also trivially satisfies condition 2 since no algebraic closure of Q contains Q(t).
I suggest rewriting the article to have condition 3 as the proper definition and detailing the non-equivalence of the other conditions for non-algebraic extensions. I realise this is a bit pedantic since usually one only deals with algebraic extensions when talking about normality but I think it's still an important technical point. Ali 24789 (talk) 02:09, 28 March 2010 (UTC)