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Math Help - Irreducible Polynomials Over A Field

  1. #1
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    Irreducible Polynomials Over A Field

    Are the polynomials in f(x) = x^3 + 2x^2 + 1 and g(x) = x^2 + x + 1 irreducible in F5[x]? Why or why not?

    I'm having trouble checking to see if they are irreducible or not. I know it has something to do with the roots but I'm not too sure. Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by Shapeshift View Post
    Are the polynomials in f(x) = x^3 + 2x^2 + 1 and g(x) = x^2 + x + 1 irreducible in F5[x]? Why or why not?

    I'm having trouble checking to see if they are irreducible or not. I know it has something to do with the roots but I'm not too sure. Any help would be appreciated.

    Hint: Over any field, a polynomial of degree less than or equal 3 (three) is irreducible iff it has no root in the field.

    Tonio
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  3. #3
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    So am I right in thinking that both f(x) and g(x) are irreducible since none of their roots = 0 (mod 5)?
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    Quote Originally Posted by Shapeshift View Post
    So am I right in thinking that both f(x) and g(x) are irreducible since none of their roots = 0 (mod 5)?

    I supose you meant "...since none of the elements of the field \mathbb{F}_5 is a root of any of them", and then yes: you'd be right.

    Tonio
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  5. #5
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    Haha sorry, I'm new to this concept so I'm still having a little trouble truly understanding it. Thank you!
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