Algebraic independence via the Jacobian

I have seen being mentioned that algebraic independence of polynomials can be tested by the so called Jacobian Criterion (Apparently one takes the Jacobian matrix of these polynomials and inspects the rank of the matrix (or the rank of its minors)). Where can i find the precise statement and its proof?

• mathoverflow.net/questions/41535/… Apr 30, 2015 at 16:30
• @user26857: I am aware of that post. However, the proof given in the comments is beyond my understanding. I was wondering if some more basic proof is possible, or if someone can translate it into commutative algebra. Apr 30, 2015 at 18:10
• My understanding was that $x^2$ and $y^2$ were algebraicly independent, even though the jacobian can be 0. What am I missing? May 11, 2021 at 18:35