transcendence degree

English

Noun

transcendence degree (plural transcendence degrees)

(algebra, field theory, of a field extension) Given 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 ${\\displaystyle F_{1}/k}$ and ${\\displaystyle F_{2}/k}$ denote algebraic function fields of transcendence degree one.
- 2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422,
  Lemma 7.19 Suppose that ${\\displaystyle L}$ is a field extension^{[sic – meaning extension field]} of transcendence degree ${\\displaystyle r}$ over a field ${\\displaystyle K}$ and that ${\\displaystyle L}$ is not separably generated over ${\\displaystyle K}$ . If ${\\displaystyle x_{1},\\dots ,x_{n}}$ are elements of ${\\displaystyle L}$ such that ${\\displaystyle L=K(x_{1},\\dots ,x_{n})}$ , then for a suitable relabeling of the ${\\displaystyle x_{i}}$ 's, the subfield ${\\displaystyle K(x_{1},\\dots ,x_{r+1})}$ of ${\\displaystyle L}$ is of transcendence degree ${\\displaystyle r}$ and is not separably generated over ${\\displaystyle K}$ .
- 2008, Bernd Sturmfels, Algorithms in Invariant Theory, Springer, 2nd Edition, page 24,
  Proposition 2.1.1 Every finite matrix group ${\\displaystyle \\Gamma \\subset \\operatorname {GL} (\\mathbb {C} ^{n})}$ has ${\\displaystyle n}$ algebraically independent invariants, i.e., the ring ${\\displaystyle \\mathbb {C} [x]^{\\Gamma }}$ has transcendence degree ${\\displaystyle n}$ over ${\\displaystyle \\mathbb {C} }$ .

Usage notes

A transcendence degree is said to be of a field extension (i.e., ${\\displaystyle L/K}$ ). More properly, it is the cardinality of a particular type of subset of the extension field ${\\displaystyle L}$ , although the context of the field extension is required to make sense of the definition.
Relatedly, a transcendence basis of ${\\displaystyle L/K}$ $\text{[math]}$ is a subset of ${\\displaystyle L}$ $\text{[math]}$ that is algebraically independent over ${\\displaystyle K}$ $\text{[math]}$ and such that ${\\displaystyle L}$ $\text{[math]}$ is an algebraic extension of ${\\displaystyle K(S)}$ $\text{[math]}$ (that is, ${\\displaystyle L/K(S)}$ $\text{[math]}$ is an algebraic extension).
- It can be shown that every field extension ${\\displaystyle L/K}$ has a transcendence basis, whose cardinality, denoted ${\\displaystyle \\operatorname {trdeg} _{K}L}$ or ${\\displaystyle \\operatorname {trdeg} (L/K)}$ , is exactly the transcendence degree of ${\\displaystyle L/K}$ .