More generally, an

element x of a

commutative ring L that has an

identity element is called algebraic over a

ring K (where L is an extension of K) with the same properties if there exists a

nonzero polynomial f with coefficients in K such that f(x)=0. The most common case is to use the natural or

rational numbers for K and the real or

complex numbers for L.

A little known (or rather, rarely realized) fact is that if K is countable, then so is the set of all its (finite) polynomials, and thus (due to the fundamental theorem of algebra) the set of all algebraic numbers over K! This means that there are in fact no more polynomial roots of any degree, as well as finite sums, products or powers (or any combination thereof) of polynomial roots than there are natural numbers. While algebraic numbers are the most commonly encountered irrational numbers, they are in fact damn rare and there are far more transcendental ones.