purely transcendental extension

A purely transcendental extension is the field-theoretic version of working with formal variables and nothing more. You adjoin a set of completely free, unrelated 'indeterminates' to your base field and take all the rational expressions you can form. There is no hidden algebra: no element satisfies any polynomial relation forced by the others. It is the cleanest, most generic way to enlarge a field, with no algebraic part on top.

Precisely, an extension L of K is purely transcendental if L = K(S) for some transcendence basis S, equivalently if L is K-isomorphic to a rational function field K(x_1, ..., x_n) (or K of countably or larger many variables in the infinite case). Here the variables are algebraically independent over K, so K[x_1, ..., x_n] is an honest polynomial ring and L is its field of fractions.

Every extension factors through a purely transcendental piece: pick a transcendence basis S, and then L is algebraic over the purely transcendental K(S). The purely transcendental part carries all the 'free dimensions' and the algebraic part carries the rest. A warning worth remembering: even when L is algebraic over a purely transcendental subfield, L itself need not be purely transcendental — Luroth's theorem rescues this only for transcendence degree 1 over the base.