if , then ;
if , then there is a finite subset of , such that ;
if is a subset of such that implies , then implies ;
if but for some , then .
Given a dependence relation on , a subset of is said to be independent if for all . If , then is said to span if for every . is said to be a basis of if is independent and spans .
Remark. If is a non-empty set with a dependence relation , then always has a basis with respect to . Furthermore, any two of have the same cardinality.
Let be a field extension of . Define by if is algebraic over . Then is a dependence relation. This is equivalent to the definition of algebraic dependence.
|Date of creation||2013-03-22 14:19:25|
|Last modified on||2013-03-22 14:19:25|
|Last modified by||CWoo (3771)|