Definition of Polynomial with infinitely many variables

Question

I've read this and curious about polynomial with infinitely many variables. Where I can find the definition of polynomial with infinitely many variables? I have to write this definition in my final assignment, thanks for your help.

Answer 1

Let $X$ be any set. The ring of polynomials in $X$ over a commutative ring $K$ is the set of functions $(\mathbb N^X)' \to K$ with finite support, that is, which are zero except for a finite subset. Here, $(\mathbb N^X)'$ is the set of functions $X \to \mathbb N$ with finite support. These functions choose a finite set of variables from $X$ and assigns degrees to them. These are the monomials. Finally, a function $(\mathbb N^X)' \to K$ assigns coefficients to monomials.

Answer 2

If $X$ is a set of indeterminates, denote the polynomial ring over this set of indeterminates as $R[X]$. (This bends the normal usage, because normally the thing inside the brackets is already the indeterminate. But it is awkward to write it other ways.)

Explicitly, an element of $R[\{x_i|i\in I\}]$ is an $R$-linear combination of elements from $\{x_i|i\in I\}$.

Here "linear combination" means implicitly "finite linear combination", that is, you take finitely many elements of $\{x_i|i\in I\}$, multiply each one with an element of $R$ if you want, then add them up. That's a linear combination.

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If you believe in polynomial rings with finitely many indeterminates, another way you could rephrase this is like this:

Then $R[\{x_i|i\in I\}]:=\bigcup\left\{R[\{x_i|i\in F\}]\mid F \text{ ranges over finite subsets of } I\right\}$