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Math Help - listing polynomials?

  1. #1
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    listing polynomials?

    hi another one guys:

    any clues on how to go about doing these questions would be greatly appreciated i will show my working once i have an idea

    (a) list all the monic polynomials over F_2 of degree  \leq 3 .

    (b) determine which of these polynomials are irreducible over F_2.

    (c) Factorise the reducible polynomials into irreducible polynomials.

    any clues appreciated guys thanks.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by smoothman View Post
    hi another one guys:

    any clues on how to go about doing these questions would be greatly appreciated i will show my working once i have an idea

    (a) list all the monic polynomials over F_2 of degree  \leq 3 .

    (b) determine which of these polynomials are irreducible over F_2.

    (c) Factorise the reducible polynomials into irreducible polynomials.

    any clues appreciated guys thanks.
    what does F_2 mean here?
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  3. #3
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    Quote Originally Posted by smoothman View Post
    hi another one guys:

    any clues on how to go about doing these questions would be greatly appreciated i will show my working once i have an idea

    (a) list all the monic polynomials over F_2 of degree  \leq 3 .

    (b) determine which of these polynomials are irreducible over F_2.

    (c) Factorise the reducible polynomials into irreducible polynomials.

    any clues appreciated guys thanks.
    To Jhevon: Here \mathbb{F}_2 is the field that has two elements (another notation \mbox{GF}(2) which means "Galois field").

    And degree <=3 polynomial is,
    Ax^3+Bx^2+Cx+D where A,B,C,D\in \mathbb{F}_2.

    We can think of this field as \mathbb{Z}_2 \left< + , \cdot \right>
    So the elements are:
    x^3,x^3+x^2,x^3+x,x^3+1,x^3+1,x^3+x^2+1,x^3+x^2+x, x^3+x^2+1.x^3+x^2+x+1,x^2,x^2+x,x^2+1,x^2+x+1,x,x+ 1

    Now to see which are irreducible over this field simply check if it has a zero or not (because it is of degree 3 or less).
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