Algebraic Dependence
Definition: If K ⊆ L is an extension of fields and a is an element of L, then we say that a is algebraic over K if there is nonzero polynomial P(X) with coefficients from K so that P(a) = 0
 
Definition: Given a subset X of a field L and K a subfield of L, we say that X is algebraically independent over K if for each element a of X, a is not algebraic over the field generated by X - {a} and K. The transcendence degree of L over K, written tr.deg(L/K), is the cardinality of any transcendence basis, a maximal algebraically independent (over K) subset of L.
 
With the definition of transcendence degree we implicitly use the fact (due to Steinitz) that any two transcendence bases have the same cardinality.