The Krull dimension one is one of the few that might not be found in an undergraduate program. You'd probably come across it in an algebra class that straddles

The Krull dimension one is one of the few that might not be found in an undergraduate program. You'd probably come across it in an algebra class that straddles the line between upper undergrad and intro graduate coursework (but Noetherian integral domain is something you would usually find in the sophomore/junior undergrad algebra class often taken as a prerequisite for the other hypothetical class). Also > "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ" I don't think that this is high school level. Secondary school level math rarely touches anything related to multivariate polynomials (and usually doesn't have a notion of "closed sets" beyond closed intervals, but that's not as relevant apart from how the problem is stated).