In mathematics, a transcendental extension L / K {\displaystyle L/K} is a field extension such that there exists an element in the field L {\displaystyle L} that is transcendental over the field K {\displaystyle K} ; that is, an element that is not a root of any univariate polynomial with coefficients in K {\displaystyle K} . In other words, a transcendental extension is a field extension that is not algebraic. For example, C {\displaystyle \mathbb {C} } and R {\displaystyle \mathbb {R} } are both transcendental extensions of Q . {\displaystyle \mathbb {Q} .}
A transcendence basis of a field extension L / K {\displaystyle L/K} (or a transcendence basis of L {\displaystyle L} over K {\displaystyle K} ) is a maximal algebraically independent subset of L {\displaystyle L} over K . {\displaystyle K.} Transcendence bases share many properties with bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero. Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic variety is the transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory in positive characteristic a role that is very similar to the role of algebraic number fields in characteristic zero.
Transcendence basis Zorn's lemma shows there exists a maximal linearly independent subset of a vector space (i.e., a basis). A similar argument with Zorn's lemma shows that, given a field extension L / K, there exists a maximal algebraically independent subset of L over K. It is then called a transcendence basis. By maximality, an algebraically independent subset S of L over K is a transcendence basis if and only if L is an algebraic extension of K(S), the field obtained by adjoining the elements of S to K. The exchange lemma (a version for algebraically independent sets) implies that if S and S' are transcendence bases, then S and S' have the same cardinality. Then the common cardinality of transcendence bases is called the transcendence degree of L over K and is denoted as t r . d e g . K L {\displaystyle \operatorname {tr.deg.} _{K}L} or t r . d e g . ( L / K ) {\displaystyle \operatorname {tr.deg.} (L/K)} . There is thus an analogy: a transcendence basis and transcendence degree, on the one hand, and a basis and dimension on the other hand. This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of finitary matroids (pregeometries). Any finitary matroid has a basis, and all bases have the same cardinality. If G is a generating set of L (i.e., L = K(G)), then a transcendence basis for L can be taken as a subset of G. Thus, t r . d e g . K L ≤ {\displaystyle \operatorname {tr.deg.} _{K}L\leq } the minimum cardinality of generating sets of L over K. In particular, a finitely generated field extension admits a finite transcendence basis. If no field K is specified, the transcendence degree of a field L is its degree relative to some fixed base field; for example, the prime field of the same characteristic, or K, if L is an algebraic function field over K. The field extension L / K is purely transcendental if there is a subset S of L that is algebraically independent over K and such that L = K(S). A separating transcendence basis of L / K is a transcendence basis S such that L is a separable algebraic extension over K(S). A field extension L / K is said to be separably generated if it admits a separating transcendence basis. If a field extension is finitely generated and it is also separably generated, then each generating set of the field extension contains a separating transcendence basis. Over a perfect field, every finitely generated field extension is separably generated; i.e., it admits a finite separating transcendence basis.
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