Results 1 to 2 of 2

Math Help - Questions about polynomials over Z mod 5

  1. #1
    Newbie
    Joined
    May 2009
    Posts
    7

    Questions about polynomials over Z mod 5

    How many irreducible quadratic polynomials are there over Z mod 5?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by xxie View Post

    How many irreducible quadratic polynomials are there over Z mod 5?

    Thanks!
    let's solve the problem for the general case, i.e. over \mathbb{Z}/p, where p is any prime number. a quadratic polynomial is irreducible iff it has no root. this number is obviously n=p^2(p-1) - k, where

    k is the number of those polynomials of degree 2 which are reducible, i.e. they are in the form a(x-b)(x-c). clearly a can be any element of \mathbb{Z}/p - \{0 \}. so there are p-1 possibilities for a.

    we also have \binom{p}{2} distinct pairs a,b in \mathbb{Z}/p. it is also possible to have a=b, which gives us p possibilities. thus k=(p-1) \left(\binom{p}{2} + p \right)=\frac{p(p^2-1)}{2}. thus: n=p^2(p-1)-k=\frac{p(p-1)^2}{2}.

    if you're looking for the number of monic irreducible polynomials of degree 2, then a=1 would be the only possibility for a and thus in this case n=\frac{p(p-1)}{2}.
    Last edited by NonCommAlg; May 6th 2009 at 09:40 PM. Reason: typo!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 2 polynomials questions
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: April 2nd 2010, 02:44 AM
  2. Polynomials questions again... part 6
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: July 15th 2009, 08:50 AM
  3. Polynomials questions again... part 5
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: July 12th 2009, 04:37 AM
  4. Polynomials questions again... part 1
    Posted in the Calculus Forum
    Replies: 5
    Last Post: June 27th 2009, 04:48 AM
  5. Polynomials questions again... part 2
    Posted in the Calculus Forum
    Replies: 5
    Last Post: June 25th 2009, 07:44 AM

Search Tags


/mathhelpforum @mathhelpforum