Transcendental numbers | Algebraic varieties | Matroid theory | Field (mathematics)

Transcendence degree

In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of L over K. 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) (the field 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 and is denoted trdegK L or trdeg(L / K). 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., the rational numbers field Q if L is of characteristic 0 and the finite field Fp if L is of characteristic p. 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). (Wikipedia).

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Related pages

Algebraic extension | Transcendental number | Linear span | Matroid rank | Vector space | Algebraic closure | Algebraic variety | Automorphism | Algebraic number | Linear independence | Function field of an algebraic variety | E (mathematical constant) | Regular extension | Empty set | Matroid | Meromorphic function | Field extension | Riemann surface | Characteristic (algebra) | Continuum hypothesis | Union (set theory) | Pi | Subset | Algebraic independence | Basis (linear algebra) | Compact space | Abstract algebra | Complex number | Cardinality | Carl Ludwig Siegel