# Can a polynomial have uncountably many indeterminates?

I am writing up some stuff on polynomials with arbitrarily many variables over/with coefficients in $$\mathbb{C}$$. While doing this, I vaguely remembered reading something about 'polynomials with uncountably many indeterminates,' or something along those lines, but can't find anything about it now that I look for it.

My question: is it sensible to say a polynomial may have 'uncountably many indeterminates'? What does this...mean? Are we not just handling some element of a polynomial ring $$K[x_1,x_2\dots x_n]$$, which obviously has countably many indeterminates?

• a polynomial ring is just a free $k$-algebra over a set $X$ (the variables), if you now choose $X$ to be uncountable you get the ring of polynomials in uncountably many variables, but it is highly nonnoetherian, and hence quite ill behaved. Dec 9 '19 at 16:18
• Yes, we have $K[x_i\mid i\in \Bbb R]$, why not? Dec 9 '19 at 16:18
• @DietrichBurde you mean $i \in \mathbb{R}$, don't you? And sorry, meant "nonnoetherian"! Dec 9 '19 at 16:20
• @Enkidu Ah, sorry, you are right. Uncountably many, not infinitely many. Dec 9 '19 at 16:23