Home > Transcendence degree

A subset S of L is a transcendence basis of L/K if it is algebraically independent over K and if furthermore L is an algebraic extension of the field K(S) obtained by adjoining the elements of S to K. One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension.

If no field K is specified, the transcendence degree of a field L is its degree relative to the prime field of the same characteristic, i.e., Q if L is of characteristic 0 and F_p if L is of characteristic p.

The field extension L/K is purely transcendental if there is a subset S of L that's algebraically independent over K and such that L = K(S).

1 Examples

• Every algebraic extension has transcendence degree 0; the empty set serves as a transcendence basis here.
• The field of rational functions in n variables K(x₁,...,x_n) is a purely transcendental extension with transcendence degree n over K; we can for example take {x₁,...,x_n} as a trancendence base
• More generally, the transcendence degree of the function field L of an n-dimensional algebraic variety over a ground field K is n.
• Q( √2, π) has transcendence degree 1 over Q because √2 is algebraic while π is transcendental.
• The transcendence degree of C or R over Q is the cardinality of the continuum.
• The transcendence degree of Q(π, e) is either 1 or 2; the precise answer is unknown because we don't know whether π and e are algebraically independent.

2 Analogy with vector space dimensions

There is an analogy with the theory of vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for dimensions. The dictionary matches algebraically independent sets with linearly independent setsAbstract algebra Algebra Linear algebra In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. For instance, in thr; sets S such that L is algebraic over K(S) with spanning setsIn the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. The linear span of a set of vectors is a therefore a vector space but unlike a basis the vectors need not be linearly; transcendence bases with basesAbstract algebra Algebra Linear algebra In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V''. B is a mi; and transcendence degree with dimension. The fact that transcendence bases always exist (like the fact that bases always exist in linear algebra) requires the axiom of choiceSet theory In mathematics, the axiom of choice is an axiom in set theory. It was formulated about a century ago by Ernst Zermelo, and was quite controversial at the time. It states the following: Stated more formally: Another formulation of the axiom of c. The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma.