I Analogy question for algebraists

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Polynomials with one variable and coefficients in the field K are to finite dimensional K vector spaces as polynomials in several variables over the field K are to ....?
An "analogy question":

Polynomials with one variable and coefficients in the field K are to finite dimensional K vector spaces as polynomials in several variables over the field K are to ....?

As a teenager, I recall taking tests that had "analogy questions" on them. The format was: Thing A is to thing B as Thing C is to ...what? They had objectively correct answers - at least in the mind of the people that posed them.

So I'm wondering if the above question has an answer that most algebraists would agree with. Thanks to Axler's book Linear Algebra Done Right, I can see that iterating a linear transformation ##T## applied to a particular vector ##v## leads to the existence of a (finite degree) polynomial in powers of ##Tv## that must equal to zero. That heads toward the topic of eigenvalues and eigenvectors. I don't know if there is some analogy to that situation that involves polynomials in several variables.
 
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... to the tensor product of as many vector spaces as there are variables.
 

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I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
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Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
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Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
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