1. The problem statement, all variables and given/known data Show that [itex] \mathbb Q[x,y]/(x^2+y^2-1) [/itex] is not a unique factorization domain. 3. The attempt at a solution We have tried a few approaches. Using  to denote equivalence classes, we note that we can write [itex] [x]^2 = [1-y][1+y][/itex]. Our goal was to show that this is a non-unique prime decomposition, but this doesn't work since neither of these elements are prime. Similarly, [itex] [x+y-1][x+y+1] = [2x][y] [/itex] but the same problem applies. Next we tried playing around with evaluations of polynomials. In particular, in the quotient ring we can write any element as [itex] p(x) + y q(x) [/itex]. Taking a product of such polynomials and evaluating at zero gives a structure similar to [itex] \mathbb C[/itex]. Another suggestion was to use Laurent polynomial. Any ideas?