Editing Matroid (section)

===Matroids from field extensions===
A third original source of matroid theory is [[field theory (mathematics)|field theory]].

An [[extension field|extension]] of a field gives rise to a matroid: 
: Suppose <math>F</math> and <math>K</math> are fields with <math>K</math> containing <math>F</math>.  Let <math>E</math> be any finite subset of <math>K</math>.
: Define a subset <math>S</math> of <math>E</math> to be [[algebraic independence|algebraically independent]] if the extension field <math>F(S)</math> has [[transcendence degree]] equal to <math>|S|</math>.<ref name=Ox215>{{harvp|Oxley|1992|p=215}}</ref>

A matroid that is equivalent to a matroid of this kind is called an [[algebraic matroid]].<ref name=Ox216>{{harvp|Oxley|1992|p=216}}</ref>  The problem of characterizing algebraic matroids is extremely difficult; little is known about it. The [[Vámos matroid]] provides an example of a matroid that is not algebraic.