Results 1 to 3 of 3

Math Help - Irreducible polynomial in field

  1. #1
    Newbie
    Joined
    Sep 2012
    From
    europe
    Posts
    2

    Irreducible polynomial in field

    I have an exam on Monday, and I am not sure about the following.

    In the field F= Z5[x] / <x^3 - x^2 - 1> is x^3 - x^2 - 1 irreducible? If not list the irreducible factors.

    I am thinking that in F every element is written as something + <x^3 - x^2 - 1> and therefore the polynomial cannot be irreducible because any factor must at least include a multiple of <x^3 - x^2 - 1> and is thus of the same degree.

    But my gut tells me that the polynomial should be irreducible....

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

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    Re: Irreducible polynomial in field

    Quote Originally Posted by idontknowlol View Post
    In the field F= Z5[x] / <x^3 - x^2 - 1> is x^3 - x^2 - 1 irreducible? If not list the irreducible factors.
    Let p = x^3 - x^2 - 1.
    Taking on faith that F is actually a field (i.e that p is irreducible in Z5[x]), your question amounts to: Is the zero element/polynomial irreducible?
    That's because in Z5[x]/(p), p = 0, or more accurately, p+(p) = 0+(p).
    This is one of those special case situations where you have to read the definition of irreducible carefully.

    I don't have a text nearby, so I'll use wikipedia:
    1) "For any field F, the ring of polynomials with coefficients in F is denoted by F[x]. A polynomial p(x) in is called irreducible over F if it is non-constant and cannot be represented as the product of two or more non-constant polynomials from F[x]."
    2) "In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units."

    In both cases, the special case of a 0 element would be excluded. Thus I read that as 0 should not called irreducible.
    Last edited by johnsomeone; September 22nd 2012 at 08:36 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2012
    From
    europe
    Posts
    2

    Re: Irreducible polynomial in field

    Great. Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: September 3rd 2012, 04:38 AM
  2. irreducible polynomials over the field of rationals
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 12th 2012, 03:18 PM
  3. 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
  4. Irreducible Polynomials Over A Field
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: January 20th 2011, 07:32 PM
  5. Splitting field of an irreducible polynomial
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: June 25th 2009, 12:14 AM

Search Tags


/mathhelpforum @mathhelpforum