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: Tonio
Or am I way off base with all of this?