Results 1 to 5 of 5

Math Help - Irreducibility in Z

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    6

    Irreducibility in Z

    Prove that polynomial  \prod_{1\leq i\leq j\leq \phi(n)}(x-\omega_n^{q_i+q_j}) is irreducible in Z where gcd(q_i, n)=1, \omega_n=e^{i\frac{2\pi}n} and \phi Euler's totient function.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    This is wrong. Let \zeta = e^{2\pi i/5}.
    Then this polynomial is:
    (x-\zeta^2)(x-\zeta^3)(x-\zeta^4)(x-\zeta^5)
    (x-\zeta^4)(x-\zeta^5)(x-\zeta^6)
    (x-\zeta^6)(x-\zeta^7)
    (x-\zeta^8)

    Using the fact that \zeta^5 = 1 we get:
    (x-1)(x-\zeta)^2(x-\zeta^2)^2(x-\zeta^3)^2(x-\zeta^4)^2.

    Use the fact that (x-\zeta)(x-\zeta^2)(x-\zeta^3)(x-\zeta^4) = x^4+x^3+x^2+x+1
    This gives:
    (x-1)(x^4+x^3+x^2+x+1)^2
    This is not irreducible.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2008
    Posts
    6
    You forgot one member (x-1) but polynomial is still not irreducible.
    Thanks!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by rodeo View Post
    You forgot one member (x-1) but polynomial is still not irreducible.
    Thanks!
    Your problem looks similar to this, hopefully you never seen it before. Let \zeta = e^{2\pi i/n}. Now define f(x) = \prod_{1\leq k\leq n}^{\gcd(n,k)=1} (x - \zeta^k).
    It turns out that f(x) \in \mathbb{Z}[x] and f(x) is irreducible.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Aug 2008
    Posts
    6
    I know for that so called cyclotomic polynomials and there are my primerily idea to creating the problem.

    Right question is :
    find S such \prod_{1\leq i<j\leq k}(x-\omega_n^{q_iq_j})\in Z[x], q_i\in S, 1\leq i\leq k.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Irreducibility
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 7th 2011, 05:37 PM
  2. Irreducibility in Zp
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 23rd 2010, 09:08 AM
  3. Irreducibility
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 5th 2009, 11:52 AM
  4. irreducibility
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 21st 2009, 09:42 PM
  5. irreducibility
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 30th 2008, 08:15 AM

Search Tags


/mathhelpforum @mathhelpforum