For a polynomial ring $F[X]$ where $F$ is a field, what is the usual convention for capital or small letters for $X$?
Q1) I.e., should it be $F[X]$ or $F[x]$?
I guess theoretically it doesn't matter, just curious to know what is the usual convention.
Q2) Also, just curious why when it comes to polynomial rings, people suddenly use capital letters to denote polynomials, e.g. $X^2+1$, instead of $x^2+1$ which is more common in other branches of math? Is there a deep reason?
This trivial question has been bugging me for some time, asking it now that I remember it.
Thanks.
The reason for the capital letters notation, as used by Bourbaki, for instance, is to insist on the fact a polynomial is not a polynomial function.
A polynomial with coefficients in a commutative ring $R$ is merely a sequence of coefficients in $R$, eventually ending in $0$s, endowed with two operations, $+$ and $\times$.
Thus the indeterminate $X$ is but the sequence $(0,1,0,0,0,\dots)$. It happens that the operations are designed so that $X^2=(0,0,1,0,0,\dots)$, $(X^3=(0,0,0,1,0,\dots)$ and so on. Even $1=(1,0,0,0,0,\dots)=X^0$.
To each polynomial $P(X)\in R[X]$ one associates a polynomial function one often denotes with the same name \begin{align*} P\colon R&\longrightarrow R\\ x&\longmapsto P(x) \end{align*} substituting the element $x$ of $R$ to each occurrence of the indeterminate $X$.
The correspondence polynomial-polynomial function is not necessarily injective, except on an infinite integral domain. For instance, in the field $\mathbf F_p$, the polynomial function associated to the non-zero polynomial $X^p-X$ is the $0$ function (this is a way of formulating Little Fermat).