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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.

RANGGAJAYA CIPTAWAN
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  • See https://en.wikipedia.org/wiki/Polynomial_ring#Infinitely_many_variables – lhf Jan 03 '20 at 15:51
  • I can't use wikipedia as a reference on my undergrad thesis. Could you tell me the title of book(s) that have that definition ? – RANGGAJAYA CIPTAWAN Jan 03 '20 at 15:56
  • @RANGGAJAYACIPTAWAN It is surprisingly hard to find a concrete citation. There isn't anything wrong with citing wikipedia as long as you correlate what you find with other sources. If you have some sort of hard requirement that you ignore what is in wikipedia, then that is unfortunate. The quality of math in wikipedia is quite good, especially for basic things. – rschwieb Jan 03 '20 at 16:24
  • Notice that the term used was "polynomial ring with infinitely many variables". Any polynomial has only a finite number of variables. – Somos Jan 03 '20 at 17:54
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    To make @Somos's claim precise, the ring $R[x_1,x_2,x_3,\ldots]$ is the union of the ascending chain $,R \subseteq R[x_1] \subseteq R[x_1,x_2] \subseteq R[x_1,x_2,x_3],\ldots$ so any polynomial in the union lies in one of the rings $R[x_1,x_2,\ldots,x_k],$ so it has has finitely many variables. Generally equational algebraic structures are closed under ascending unions. – Bill Dubuque Jan 03 '20 at 20:57
  • Regarding my prior sentence, you may find of interest more generally this paper: Anderson; Dobbs; and Zafrullah: Some applications of Zorn's lemma in algebra. – Bill Dubuque Jan 03 '20 at 21:07

2 Answers2

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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.

lhf
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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.

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

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

rschwieb
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