Meaning of transcendence degree | Babel Free

Definitions

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

Examples

“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.”

“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.”

“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.”

CEFR level