transcendence degree

plural transcendence degrees

1. field theoryalgebraGiven a field extension L / K, the largest cardinality of an algebraically independent subset of L over K.

   2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, LNCS 3076, page 263, Let F_1/k and F_2/k denote algebraic function fields of transcendence degree one.

   2 more examples

   Lemma 7.19 Suppose that L is a field extension^([sic – meaning extension field]) of transcendence degree r over a field K and that L is not separably generated over K. If x#95;1,#92;dots,x#95;n are elements of L such that L#61;K(x#95;1,#92;dots,x#95;n), then for a suitable relabeling of the x#95;i's, the subfield K(x#95;1,#92;dots,x#95;#123;r#43;1#125;) of L is of transcendence degree r and is not separably generated over K.

   2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422

   Proposition 2.1.1 Every finite matrix group #92;Gamma#92;subset#92;operatorname#123;GL#125;(#92;Cⁿ) has n algebraically independent invariants, i.e., the ring #92;C#91;x#93;#92;Gamma has transcendence degree n over #92;C.

   2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24

   Synonyms: transcendental degree