Talk:Gauss's Lemma (Polynomial Theory)
Notes to self/future editors, since I don't think I'll have time to make the changes this evening (can't remember if I started this page or not, the mistakes could be mine):
- the statement is false in general if $\deg h = 0$. Statement is updated, but there must be details in the sufficient condition part of the proof missing, since presumably this fact is not used
- necessary condition isn't quite so obvious, cf. notion of an inert embedding of rings (as defined by Cohn, Algebra) --Linus44 (talk) 19:42, 8 March 2013 (UTC)
- It wasn't your page, worry not. This is your first time to touch it. Feel free to take it on whenever you like. --prime mover (talk) 19:49, 8 March 2013 (UTC)
The altered structure is needed: the classical Gauss' lemma doesn't follow from the statement that was there, whereas this does follow from classical Gauss' lemma. Plus this works better with the "Properties of Content" format. It's be nice to give corollary 2 a separate page; but I'm unable to come up with a title that doesn't seem inappropriately long. Will add in proofs soon enough and refactor. Probably today. All that remains is to add the missing proofs. --Linus44 (talk) 14:12, 14 March 2013 (UTC)
Reorganizing
Gauss' lemma on polyomials may refer to any of the following 4:
- product of primitive is primitive (valid in any ring)
- content is multiplicative (Z, UFD's, Dedekind domains, their fraction fields)
- irreducibility of polynomials in UFD's
- UFD[X] remains a UFD (now Gauss's Lemma (Ring Theory))
where 2. has an extra proof in the special case of number fields, and isn't always a trivial corollary of 1. (as it is for UFD's). All this is to say that the structure has to change from 1+corollaries to 1.2.3.(4.) next to each other. --barto (talk) (contribs) 12:43, 14 January 2018 (EST)