1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Irreducible polynomials over different fields

  1. Dec 16, 2012 #1
    Problem: Let f be monic irreducible polynomial over a field F, k be monic irreducible polynomial over a field K, deg f = deg k. Let u be common root of f and k. Prove (or disprove by counterexample), that f=k over field (F intersection K), i.e. polynomials f and k are identical.

    Proof would be easy, if polynomial f would be irreducible over some field which includes fields F and K (with using "Abel's lemma", i.e. for any element u which is algebraic over some field, there is exactly one monic irreducible polynomial f over this field, f(u)=0).
    Unfortunately, polynomial f could be reducible over smallest field which includes F and K.
    (x^2-6 is irreducible over Q(sqrt(2)) and Q(sqrt(3)) but reducible over Q(sqrt(2), sqrt(3)).

    On the other side, if polynomial f-k would be nontrivial, deg (f-k) < deg f=deg k, f-k has u as root. But, we cannot use Abels lemma again, because of polynomial f-k is over larger field than F and K a so can be reducible over this field. It doesn't follow f=k.

    How make the right proof (or find counterexample - but I think, that claim is true, i.e. f=k)?
     
    Last edited: Dec 16, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Irreducible polynomials over different fields
  1. Irreducible polynomial (Replies: 0)

Loading...