Transcendence basis #
This file defines the transcendence basis as a maximal algebraically independent subset.
Main results #
exists_isTranscendenceBasis
: a ring extension has a transcendence basisIsTranscendenceBasis.lift_cardinalMk_eq
: any two transcendence bases of a domain have the same cardinality.
References #
TODO #
Define the transcendence degree and show it is independent of the choice of a transcendence basis.
Tags #
transcendence basis, transcendence degree, transcendence
Type
version of exists_isTranscendenceBasis
.
If x
is a transcendence basis of A/R
, then it is empty if and only if
A/R
is algebraic.
If x
is a transcendence basis of A/R
, then it is not empty if and only if
A/R
is transcendental.
If R
is a commutative ring and A
is a commutative R
-algebra with injective algebra map
and no zero-divisors, then the R
-algebraic independent subsets of A
form a matroid.
Equations
- AlgebraicIndependent.matroid R A = (AlgebraicIndependent.indepMatroid✝ R A).matroid.copyBase Set.univ (fun (s : Set A) => IsTranscendenceBasis R Subtype.val) ⋯ ⋯
Instances For
If s ⊆ t
are subsets in an R
-algebra A
such that s
is algebraically independent over
R
, and A
is algebraic over the R
-algebra generated by t
, then there is a transcendence
basis of A
over R
between s
and t
, provided that A
is a domain.
This may fail if only R
is assumed to be a domain but A
is not, because of failure of
transitivity of algebraicity: there may exist a : A
such that S := R[a]
is algebraic over
R
and A
is algebraic over S
, but A
nonetheless contains a transcendental element over R
.
The only R
-algebraically independent subset of {a}
is ∅
, which is not a transcendence basis.
See the docstring of IsAlgebraic.restrictScalars_of_isIntegral
for an example.
Any two transcendence bases of a domain A
have the same cardinality.
May fail if A
is not a domain; see https://mathoverflow.net/a/144580.