Talk:Splitting field

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated Start-class, Mid-priority)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
Start Class
Mid Priority
 Field: Algebra

from Splitting Feild[edit]

Splitting Feild : Definition[edit]

          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 ]

Rich Farmbrough 11:35, 17 October 2005 (UTC)

Equation for L: The extension has degree 6, do not introduce redundant terms.[edit]

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 . 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)

Merging content from Construction of splitting fields[edit]

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. -- (talk) 01:42, 11 June 2009 (UTC)

I agree, but this discussion seems to be stale. Who's going to do it? Marc van Leeuwen (talk) 12:46, 3 April 2011 (UTC)
I also agree and I will do it. Bender2k14 (talk) 20:21, 17 May 2011 (UTC)
Done. Bender2k14 (talk) 21:52, 17 May 2011 (UTC)