/sci/ - Science & Math

>>12573717
Think of a polynomial in one variable as a sequence. Let's say we have [math]f\in R[x][/math] for some ring [math]R[/math]. Then [math]f = (a_n)_{n \ge 0}[/math], where each [math]a_n \in R[/math] is the coefficient of the nth power of our variable. Furthermore, there is a maximal index for non-zero coefficients, so you are essentially talking about the set of those integer sequences that have only a finite number of non-zero entries. I think, yet I'm not completely sure, that you could then take all the primes, and then order them somehow (let's just go with the natural order), and then define a function [math]R[x] \to \mathbb{Q}[/math] by [math](a_n)_n \mapsto \prod p_{n+1}^{a_n}[/math], (n starts from 0), and this should be a well-defined injection. If this was the case, then it would be a countable set. Err... Exercise of the night: prove or disprove my post.