Talk:Splitting field
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[edit] from Splitting Feild
[edit] Splitting Feild : Definition
Let F be any field, and f be a monic polynomial of degree n in F[X]. This polynomial is said to split in F if it factors completely, i.e., factors as a product of n linear factors x-ri. The ri are then the roots of f, that is, the solutions of the equation f(x)=0. If K is some extension of F, we likewise say f splits in K if can be written as a product (x-r_1)(x-r_2)...(x-r_n) of n linear factors in K[X].
Clearly f then splits also in F(r_1,r_2,...,r_n), the subfield of K generated by the roots. We say that K is a splitting field of f over F if f splits in K and K=F(r_1,r_2,...,r_n).
See also: [ Construction Of splitting Feilds http://en.wikipedia.org/wiki/Construction_of_splitting_fields ]
Rich Farmbrough 11:35, 17 October 2005 (UTC)
