SOLUTION: factor x^3-y^3 into irreducibles in Q(x,y) and prove that each of the factors is irreducible.

Question 557105: factor x^3-y^3 into irreducibles in Q(x,y) and prove that each of the factors is irreducible.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!

x-y is obviously irreducible. For sake of completeness, we show that x^2 + xy + y^2 is irreducible. Suppose that, on the contrary, it is reducible into two polynomials. Then we can write it in a general form:

Hence by equating coefficients we have ac = 1, ad + bc = 0, a+c = 0, bd = 0. Since bd = 0, one of b or d must be 0. Assume that b = 0 (we can do this due to symmetry). Then ad = 0. Since ac = 1, a cannot equal 0 so d = 0. Now we have ac = 1 and a+c = 0. This does not reveal any solutions in real numbers (since a and c are additive inverses, product is negative) so we have a contradiction and x^2 + xy + y^2 is irreducible.
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