Let $k$ be a field and $f,g \in k[x,y]$ be two non-constant polynomials in two variables. Is it true that the following conditions are equivalent:

*$f$ and $g$ are relatively prime in $k[x,y]$*, in the sense that they have no non-constant common divisors, so if we have the identity $\alpha f + \beta g = 0$ in $k[x,y]$ for some polynomials $\alpha,\beta\in k[x,y]$, then $\alpha = -g \gamma$ and $\beta = f \gamma$ for some $\gamma \in k[x,y]$.*The ideal $(f,g)$ generated by $f,g$ is of finite codimension in $k[x,y]$ over $k$*, i.e. $\dim_{k} k[x,y] / (f,g) < \infty$.

The following simple arguments prove that 2) implies 1).

Suppose $f = a q$ and $g = b q$ for some $a,b,q\in k[x,y]$ and $q$ is non-constant, so 1) fail. Exchanging if necessary $x$ and $y$, one can assume that $q(x,y) \not= x^k$ for all $k$. Then neither of the following infinite and linearly independent over $k$ family of polynomials $\{x^i\}_{i\geq0}$ belongs to $(q) \supset (f,g)$. Whence $\dim_{k} k[x,y] / (f,g) \geq \dim_{k} k[x,y] / (q) = \infty$, and thus 2) fails as well.

Thus the question is whether 1) implies 2)?

Perhaps one should assume that $k$ has characteristic $0$.

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