Results 1 to 2 of 2

Math Help - Minimal polynomial of a matrix over a field F and over an extension of F.

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    138

    Minimal polynomial of a matrix over a field F and over an extension of F.

    Let F and K be fields, F subfield of K. Given a matrix T in Fn (the algebra of n x n matrices over F), let p be the minimal polynomial of T over F and p0 the minimal polynomial of T over K, considering T as an element of Kn. Then p0 | p, since p0 divides all polynomials over K (and hence all polynomials over F) which are satisfied by T. Of course p0 | p in K[x] but not necessarily in F[x]. My question is: how could I find a matrix T in Fn such that p0 is different from p?

    For example, let's make F = Q, the field of rational numbers, and K = F( \sqrt 2), the field obtained by adjoining \sqrt 2 to F (within say the field of real numbers. With n = 2, if T = (a_ij) is the matrix with a11= 1, a12= 1/2, a21= 2, a22= -1, then p= x^2 - 2, whose roots are all in K. But then p0 = p. Perhaps with n a little larger. And perhaps, after all, p0 is always equal to p.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Feb 2009
    Posts
    138

    Re: Minimal polynomial of a matrix over a field F and over an extension of F.

    Sorry, I have found that always p0 = p.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding minimal polynomial of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: May 3rd 2013, 07:51 AM
  2. roots of minimal polynomial in a simple field extension.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 1st 2013, 07:24 AM
  3. Replies: 5
    Last Post: September 4th 2012, 12:37 AM
  4. Field extension to find roots of polynomial
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: June 13th 2011, 11:22 AM
  5. Is Simple Extension Minimal?
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: May 23rd 2006, 02:17 PM

Search Tags


/mathhelpforum @mathhelpforum