transcendence degree

If the degree of an extension counts vector-space dimensions, the transcendence degree counts something coarser and more fundamental: how many independent free variables an extension secretly contains. It ignores all the algebraic 'fine structure' and reports only the number of genuinely transcendental directions. A degree-37 algebraic extension and the base field both have transcendence degree 0; the rational function field in three variables has transcendence degree 3.

Precisely, the transcendence degree of an extension L over K, written trdeg_K(L), is the cardinality of any transcendence basis of L over K. This is well defined because all transcendence bases have the same cardinality, by an exchange argument modeled on the proof that vector-space dimension is well defined. Transcendence degree 0 means exactly that L is algebraic over K.

Transcendence degree is additive in towers: trdeg_K(L) = trdeg_K(M) + trdeg_M(L) for K subset of M subset of L, the multiplicative tower law's logarithmic cousin. It is the algebraic skeleton behind the geometric dimension of a variety: the dimension of an irreducible variety equals the transcendence degree of its function field over the ground field.