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Is there a reason, why the set of polynomials of, say, a field is sometimes written as $K[X]$ and sometimes as $K[x]$, i.e. is there a reason, why the "indeterminate" is sometimes denoted with $x$ and sometimes with $X$ ?

(I'm thinking, perhaps in some areas of math/for some fields, $x$ is preferred to $X$ to maybe pronounce the fact that the formal polynomials $\sum_i a_i x^i$ correspond one-to-one to functions $f(x)=\sum_i a_i x^i$ (since if the indeterminate is denoted with $x$, there is no notational difference to the notation of a function), whereas the notation $\sum_i a_i X^i$ would maybe suggest that this correspondence isn't one-one ? )

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I think it is just notation. Often, $X$ is used to denote a term with more than one variable. – Rankeya Oct 26 '12 at 16:51
I suspect $X$ is often used when $x$ is frequently thought of as a "variable," and hence making it an indeterminate can be confusing. – Thomas Andrews Oct 26 '12 at 17:03
Isn't the corect spelling "indeterminate"? – Michael Hardy Oct 26 '12 at 17:12
@MichaelHardy Oops, yes! – temo Oct 26 '12 at 17:23
up vote 3 down vote accepted

What you suggest may certainly be a reason why some people choose one over the other, but a lot of the time I think it comes down to the fact that people like to use $x$ as an element of some set and using $X$ for an indeterminate keeps things free of confusion. For example if we want to talk about field extensions, $K(x)$ would be the field extension generated by $x$ with coefficients in K. We may want to talk about $K[X]$ in the same context, and compare the two in some way (for example by considering the map from $K[X]$ to $K(x)$ that evaluates a given polynomial at $x$. Admittedly we have different kinds of brackets to do this, but when written down the difference between $X$ and $x$ is often a lot easier to notice.

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Sometimes I see people use $K[X]$ when they plan to use $x$ as an algebraic number for some extension $K(x)$. Most often I see this in cryptography, where you'll have elements of $\mathbb{F}_{p^n}$ written as polynomials in $x$ (after which you'll deal with systems of polynomials in $\mathbb{F}_{p^n}[X]$).

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