Results 1 to 3 of 3

Math Help - Irreducible polynomials

  1. #1
    Member
    Joined
    Jun 2008
    Posts
    148

    Irreducible polynomials

    Suppose g(x) is an irreducible in R. If f(x) is any other polynomial, does that imply that g(f(x)) is also irreducible?

    I'm particularily interested in the case when g(x) = x^2 + c.

    Here are some examples I've tried

    g(x) = x^2 + d

    1) f(x) = c

    g(f(x)) = c^2  + d is a constant

    2) f(x) = x + c

    g(f(x)) = x^{2} + 2c \cdot x + c^2 + d

    So if I choose (c,d) such that the above has integer roots then I've solved the problem.

    So I need 4c^2 - 4 \cdot (c^2 + d) to be a square. But 4 divides this, so I just need for c^2 - c^2 - d = -d to be a square.

    This is a contradiction to the condition that g(x) = x^2 + d be irreducible because then g(x) would then be a difference of squares.

    I stopped after this......Naturally the next f(x) to try would be a quadratic, but I don't know enough about quartic polynomials to know the conditions for when they're irreducible or not.
    Last edited by jamix; March 28th 2011 at 11:13 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    I assume from the way that the problem is phrased that the scalars here are the real numbers. If so, let g(x) = x^2+1 (irreducible) and let f(x) = x^2. Then g(f(x)) = x^4+1 = (x^2+\sqrt2x+1)(x^2-\sqrt2x+1) (reducible).

    With a bit more effort, you can get a similar example with rational scalars.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2008
    Posts
    148
    Thanks. Thats a good start and got me thinking along similar lines.

    Using g(x) = x^2 + 1 and f(x) = x^d where d is odd. would give a polynomial x^{2d} + 1 = (x^2 + 1) \cdot (x^{2d-2}  -  x^{2d-4} + x^{2d-6} - ..... + 1).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Irreducible Polynomials f(x) = x^7 - x
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 24th 2011, 11:47 PM
  2. Irreducible polynomials in Zp
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: September 5th 2010, 03:59 PM
  3. Replies: 7
    Last Post: January 8th 2010, 04:13 AM
  4. irreducible polynomials
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 17th 2009, 03:26 PM
  5. irreducible polynomials
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 5th 2008, 03:44 AM

Search Tags


/mathhelpforum @mathhelpforum