Science Fair Project Encyclopedia
Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X].
The following conditions are equivalent to L/K being a normal extension:
- Let Ka an algebraic closure of K containing L. Every embedding σ of L in Ka such that σ restricts to the identity on K, verifies σ(L)=L. In other words, σ is an automophism of L over K.
- Every irreducible polynomial in K[X] which has a root in L factors into linear factors in L[X].
For example, Q(√2)/Q is a normal extension, but Q(4√2)/Q is not a normal extension since it is missing some roots of X4-2.
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details