§ Why division algorithm with multiple variables go bad

• In C[x, y], defining division is complicated, and needs grobner bases to work.
• It's because they don't obey the GCD property. Just because gcd(a, b) = g does not mean that there exist k, l such that ak + bl = g
• For example, in C[x, y], we have gcd(x, y) = 1 but we don't have polynomials k, l such that kx + ly = 1.
• Proof: suppose for contradiction that there do exist k, l such that kx + ly = 1. Modulo x, this means that ly = 1 which is absurd, and similarly modulo x it means kx = 1 which is also absurd.