Talk:Splitting field

This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Mathematics Mid‑priority 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.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Mid This article has been rated as Mid-priority on the project's priority scale.

Text or other creative content from this version of Construction of splitting fields was copied or moved into Splitting field with this edit on May 17, 2011. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists.

          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)Reply

Dear User:EmilJ: Thanks for your editing. Can you, please, explain me the following:

You removed ω₃ from this equation but left 2^2/3. I think that only terms a, b, c and d should be included.

TomyDuby (talk) 16:36, 28 November 2008 (UTC)Reply

No, because the extension has degree 6, not 4. ω₃ is a rational linear combination of the other basis elements, namely ${\displaystyle \omega _{3}=-1-\omega _{2}}$. In contrast to this, 2^2/3 cannot be written as a linear combination of the other basis elements. — Emil J. 10:54, 1 December 2008 (UTC)Reply

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)Reply

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)Reply

I also agree and I will do it. Bender2k14 (talk) 20:21, 17 May 2011 (UTC)Reply

Done. Bender2k14 (talk) 21:52, 17 May 2011 (UTC)Reply

I think the definition is not correct. As the polynomial p is arbitrary over K, the coefficient of the highest power is not necessarily equal to 1.Madyno (talk) 19:15, 31 March 2018 (UTC)Reply