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.