algebraic extension

Some new numbers you add to a field are “tame”: each of them is the solution of some honest polynomial equation written with coefficients from the old field. sqrt(2) is tame because it solves x^2 - 2 = 0. An algebraic extension is one where every single new element is tame in this sense — nothing wild slips in.

Precisely, an extension L/K is algebraic if every element of L is a root of some nonzero polynomial with coefficients in K. Such an element is called algebraic over K. The opposite is a transcendental element, which satisfies no polynomial over K at all. So “algebraic extension” means: no transcendental elements appear; everything you added is pinned down by an equation.

Every finite extension (one that is finite-dimensional as a vector space over K) is automatically algebraic — there simply is not enough room for a transcendental element. The converse fails: an algebraic extension can be infinite-dimensional, for instance the field of all algebraic numbers over the rationals.