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Math Help - Irreducible polynomial in two variables

  1. #1
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    Irreducible polynomial in two variables

    I'm supposed to factor (x^4)-(y^4) into a product of irreducible elements over both the fields Q[x,y] and C[x,y].

    So I've factored it into (x^2+y^2)(x+y)(x-y).

    Clearly the latter two factors are irreducible over both fields as they are linear (right?).

    But I'm stumped by the (x^2+y^2) term. I can tell that it's irreducible over Q[x,y] because it its roots are in the complex numbers, but how can I show it's irreducible over C?

    Or am I way off base with all of this?
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  2. #2
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    Quote Originally Posted by brisbane View Post
    I'm supposed to factor (x^4)-(y^4) into a product of irreducible elements over both the fields Q[x,y] and C[x,y].

    So I've factored it into (x^2+y^2)(x+y)(x-y).

    Clearly the latter two factors are irreducible over both fields as they are linear (right?).

    But I'm stumped by the (x^2+y^2) term. I can tell that it's irreducible over Q[x,y] because it its roots are in the complex numbers, but how can I show it's irreducible over C?

    You can't because it is reducible: x^2+y^2=(x-iy)(x+iy)

    Tonio

    Or am I way off base with all of this?
    .
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