Editing Transcendental extension (section)

== The transcendence degree of an integral domain ==
Let <math>A \subseteq B</math> be [[integral domain]]s. If <math>Q(A)</math> and <math>Q(B)</math> denote the fields of fractions of {{math|''A''}} and {{math|''B''}}, then the ''transcendence degree'' of {{math|''B''}} over {{math|''A''}} is defined as the transcendence degree of the field extension <math>Q(B)/Q(A).</math>

The [[Noether normalization lemma]] implies that if {{math|''R''}} is an integral domain that is a [[finitely generated algebra]] over a field {{mvar|k}}, then the [[Krull dimension]] of {{math|''R''}} is the transcendence degree of {{math|''R''}} over {{math|''k''}}.

This has the following geometric interpretation: if {{math|''X''}} is an [[affine algebraic variety]] over a field {{math|''k''}}, the Krull dimension of its [[coordinate ring]] equals the transcendence degree of its [[function field of an algebraic variety|function field]], and this defines the [[dimension of an algebraic variety|dimension]] of {{math|''X''}}. It follows that, if {{mvar|X}} is not an affine variety, its dimension (defined as the transcendence degree of its function field) can also be defined ''locally'' as the Krull dimension of the coordinate ring of the restriction of the variety to an open affine subset.