Solution Set 11 Daniel

1 (a) Claim: R[X, Y ] is not a Euclidean domain. Proof: If R[X, Y ] was a Euclidean domain, then it would be a principal ideal domain. But consider I = (x, y) = (x) + (y). I is certainly not a principal ideal, as if it were, there would be some polynomial, g that divides everything in I, so g would divide x, implying that g does not contain any monomials with positive degree in y, and likewise g would divide y, implying that g does not contain any monomials with positive degree in x, so g is a constant polynomial, contradicting the fact that I is clearly not the unit ideal.