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)
[edit] Equation for L: The extension has degree 6, do not introduce redundant terms.
Dear User:EmilJ: Thanks for your editing. Can you, please, explain me the following:
You removed ω3 from this equation but left 22/3. I think that only terms a, b, c and d should be included.
TomyDuby (talk) 16:36, 28 November 2008 (UTC)
- No, because the extension has degree 6, not 4. ω3 is a rational linear combination of the other basis elements, namely ω3 = − 1 − ω2. In contrast to this, 22/3 cannot be written as a linear combination of the other basis elements. — Emil J. 10:54, 1 December 2008 (UTC)
[edit] Merging content from Construction of splitting fields
Seems to make sense. Rather than having two separate articles, both of which begin by defining the same concept, it seems logical to me that the "Construction" article be merged into this article, as its own section to begin with. --70.53.193.251 (talk) 01:42, 11 June 2009 (UTC)
