Results 1 to 2 of 2

Math Help - Polynomial Over a Field

  1. #1
    Newbie
    Joined
    Nov 2009
    From
    Pluto
    Posts
    12

    Polynomial Over a Field

    Hi, I was preparing for my final and I was working a practice question and got very lost. Here's the problem.

    Suppose F = Z_5, and F is a field since 5 is prime.

    i.) Show the polynomial x^4+1 is irreducible over the field.
    ii.) Show that the principal ideal <x^4+1> of F[x] generated by x^4+1 is maximal and thus L=F[x]/<x^4+1> is a field.
    iii.) How many elements are in L? We can use the Euclidean algorithm (division algorithm) to show that for any g(x) in F[x], that
    g(x) + <x^4+1>=(ax+b) + <x^4+1>, where x, b are in Z_5.
    iv.) Produce addition and Cayley table.

    Attempt at the solution.

    i.) This part was pretty easy. I showed. Z_5 = {0,1,2,3,4}.
    p(x) = x^4+1
    so, p(0) = 1
    p(1) = 2
    p(2) = 17 = 2
    p(3) = 82 = 2
    p(4) = 257 = 2 (I think that calculation is right).
    But overall, this polynomial is irreducible over the Field because for each of the values there is no zero value.

    ii.) This part of the problem I have no idea where to start. In order for an ideal to be maximal, then the ideal, we'll call M is not a proper subset of any ideal except R itself. There's another definition as well, but not sure how to type it....but here goes.

    An ideal M is a maximal ideal of R iff M is a subset of I is a subset of R. (If you're not sure what I am saying here feel free to ignore it.)

    If you can help me with ii then I think I can get a better handle on the next two, but if you're feeling generous feel free to help with all . But really, it's mainly the second one.

    Thanks!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    For i) you still have to prove that the polynomial is not a product of two quadratic polynomials.

    For ii) use that if p \in R is irreducible then \langle p \rangle (the ideal generated by this element) is prime and if R is a principal ideal domain then every prime ideal is maximal.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Field extension to find roots of polynomial
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: June 13th 2011, 10:22 AM
  2. Splitting Field of a Polynomial over a Finite Field
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 1st 2011, 03:45 PM
  3. Splitting field of an irreducible polynomial
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: June 25th 2009, 12:14 AM
  4. polynomial in finite field
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 12th 2008, 08:48 AM
  5. Polynomial over a field
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 25th 2008, 11:40 AM

Search Tags


/mathhelpforum @mathhelpforum