Formal definition: Let $K$ be a fixed field. Then $$R:=\{(a_i)_{i\in\mathbb N}: a_i\in K, a_i = 0\text{ for almost all } i\in\mathbb N_0\}$$ together with the usual $+$ and $\cdot$ is a field, called the polynomial field.
Idea: A polynomial is an expression of the form $a_n x^n +a_{n-1} x^{n-1}+\dots a_2 x^2+ a_1x + a_0$, where $x$ denotes a variable and the coefficients $a_i$ are elements of $K$. For example $4x^3 + 8x^2 - x + 2$ is a polynomial (if $K=\mathbb R$). One can think of a sequence $(a_i)_{i\in\mathbb N_0} \in R$ as representing the polynomial where $a_i$ is the coefficient in front of $x^i$. Thus our example from above $4x^3 + 8x^2 - x + 2$ is represented by the sequence $(2, -1, 8, 4, 0, 0, \dots)$.
Question: I've often seen that people say "let $x$ be a symbol" in the formal definition and go on saying that they denote the set that I call $R$ by $K[x]$. I don't understand this, because in the formal definition there does never occur any variable $x$! Is the notation $K[x]$ chosen just for psychological reasons (like: "we think of the elements as being polynomial expressions that contain variables")? Or is $x$ treated as a parameter in the definition: "for every given field $K$ and every symbol $x$ we define $K[x]$ to be …". But in this case: what is a "variable"? and why doesn't the definition of $K[x]$ contain an $x$?
