Results 1 to 2 of 2

Math Help - Factor x^16+x as a product of irreducible polynomials

  1. #1
    Junior Member
    Joined
    Oct 2012
    From
    Ohio
    Posts
    48

    Factor x^16+x as a product of irreducible polynomials

    Factor x^16+x as a product of irreducible polynomials in Z2[x].

    Any help please??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,155
    Thanks
    595

    Re: Factor x^16+x as a product of irreducible polynomials

    well, ONE factor is obvious: x.

    x16 + x = x(x15 + 1).

    so now we just need to factor x15 + 1 in Z2[x].

    note that x15 + 1 = (x3 + 1)(x12 + x9 + x6 + x3 + 1).

    we can factor x3 + 1 as (x + 1)(x2 + x + 1) <---two irreducible factors (why?)

    so now we just have to factor:x12 + x9 + x6 + x3 + 1.

    note that: x15 + 1 also factors as (x5 + 1)(x10 + x5 + 1).

    since x2 + x + 1 is an irreducible factor, it must divide either x5 + 1 or x10 + x5 + 1. prove it doesn't divide x5 + 1.

    we can further factor x5 + 1 as (x + 1)(x4 + x3 + x2 + x + 1).

    prove that the quartic is irreducible (hint: it suffices to show it has no linear or quadratic factors).

    use long division to determine the quotient (xx10 + x5 + 1)/(x2 + x + 1).

    argue that since GF(16) is every root of x16 - x, and GF(8) is a subfield of GF(16), there must be an irreducible factor of x16 - x of degree 8 (we've already found the irreducible factors of orders 1,2 and 4). why does this mean we're done?
    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, 10:47 PM
  2. Replies: 7
    Last Post: January 8th 2010, 03:13 AM
  3. irreducible polynomials
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 25th 2008, 12:35 PM
  4. Product of irreducible polynomials.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 26th 2007, 06:10 PM
  5. Irreducible polynomials
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 6th 2006, 06:31 AM

Search Tags


/mathhelpforum @mathhelpforum