## ยง 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.