Field extension that is not algebraic
In mathematics, a transcendental extension is a field extension such that there exists an element in the field that is
transcendental
over the field
; that is, an element that is not a root of any
univariate polynomial
with coefficients in
. In other words, a transcendental extension is a field extension that is not
algebraic. For example,
and
are both transcendental extensions of
A transcendence basis of a field extension (or a transcendence basis of over ) is a maximal
algebraically independent
subset of
over
Transcendence bases share many properties with
bases of
vector spaces. In particular, all transcendence bases of a field extension have the same
cardinality, called the
transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero.
Transcendental extensions are widely used in
in characteristic zero.
Transcendence basis
adjoining
the elements of
S to
K.
The
exchange lemma (a version for algebraically independent sets
[2]) implies that if
S and
S' are transcendence bases, then
S and
S' have the same
cardinality. Then the common cardinality of transcendence bases is called the
transcendence degree of
L over
K and is denoted as
or
. There is thus an analogy: a transcendence basis and transcendence degree, on the one hand, and a basis and dimension on the other hand. This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of
finitary matroids (
pregeometries). Any finitary matroid has a basis, and all bases have the same cardinality.
[3]
If G is a generating set of L (i.e., L = K(G)), then a transcendence basis for L can be taken as a subset of G. In particular, the minimum cardinality of generating sets of L over K. In particular, a
finitely generated field extension
admits a finite transcendence basis.
If no field K is specified, the transcendence degree of a field L is its degree relative to some fixed base field; for example, the
over
K.
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).
A separating transcendence basis of L / K is a transcendence basis S such that L is a
separable algebraic extension over
K(
S). A field extension
L /
K is said to be
separably generated if it admits a separating transcendence basis.
[4] If a field extension is finitely generated and it is also separably generated, then each generating set of the field extension contains a separating transcendence basis.
[5] Over a
perfect field, every finitely generated field extension is separably generated; i.e., it admits a finite separating transcendence basis.
[6]
Examples
- An extension is algebraic if and only if its transcendence degree is 0; the empty set serves as a transcendence basis here.
- The field of rational functions in n variables K(x1,...,xn) (i.e. the field of fractions of the polynomial ring K[x1,...,xn]) is a purely transcendental extension with transcendence degree n over K; we can for example take {x1,...,xn} as a transcendence base.
- More generally, the transcendence degree of the function field L of an n-dimensional algebraic variety over a ground field K is n.
- Q(.
- The transcendence degree of C or R over Q is the cardinality of the continuum. (Since Q is countable, the field Q(S) will have the same cardinality as S for any infinite set S, and any algebraic extension of Q(S) will have the same cardinality again.)
- The transcendence degree of Q(e, π) over Q is either 1 or 2; the precise answer is unknown because it is not known whether e and π are algebraically independent.
- If S is a compact Riemann surface, the field C(S) of meromorphic functions on S has transcendence degree 1 over C.
Facts
If M / L and L / K are field extensions, then
- trdeg(M / K) = trdeg(M / L) + trdeg(L / K)
This is proven by showing that a transcendence basis of M / K can be obtained by taking the union of a transcendence basis of M / L and one of L / K.
If the set S is algebraically independent over K, then the field K(S) is
isomorphic
to the field of rational functions over
K in a set of variables of the same cardinality as
S. Each such rational function is a fraction of two polynomials in finitely many of those variables, with coefficients in
K.
Two algebraically closed fields are isomorphic if and only if they have the same characteristic and the same transcendence degree over their prime field.[7]
The transcendence degree of an integral domain
Let be integral domains. If and denote the fields of fractions of A and B, then the transcendence degree of B over A is defined as the transcendence degree of the field extension
The Noether normalization lemma implies that if R is an integral domain that is a finitely generated algebra over a field k, then the Krull dimension of R is the transcendence degree of R over k.
This has the following geometric interpretation: if X is an
of
X. It follows that, if
X is not an affine variety, its dimension (defined as the transcendence degree of its function field) can also be defined
locally as the Krull dimension of the coordinate ring of the restriction of the variety to an open affine subset.
Relations to differentials
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Let be a finitely generated field extension. Then[8]
where denotes the module of
Kahler differentials
. Also, in the above, the equality holds if and only if
K is separably generated over
k (meaning it admits a separating transcendence basis).
Applications
Transcendence bases are useful for proving various existence statements about
field automorphism
f of
K, there exists a field automorphism of
L which extends
f (i.e. whose restriction to
K is
f). For the proof, one starts with a transcendence basis
S of
L /
K. The elements of
K(
S) are just quotients of polynomials in elements of
S with coefficients in
K; therefore the automorphism
f can be extended to one of
K(
S) by sending every element of
S to itself. The field
L is the
algebraic closure of
K(
S) and algebraic closures are unique up to isomorphism; this means that the automorphism can be further extended from
K(
S) to
L.
As another application, we show that there are (many) proper subfields of the
surjective
. Any such map can be extended to a field homomorphism
Q(
S) →
Q(
S) which is not surjective. Such a field homomorphism can in turn be extended to the algebraic closure
C, and the resulting field homomorphisms
C →
C are not surjective.
The transcendence degree can give an intuitive understanding of the size of a field. For instance, a theorem due to Siegel states that if X is a compact, connected, complex manifold of dimension n and K(X) denotes the field of (globally defined) meromorphic functions on it, then trdegC(K(X)) ≤ n.
See also
References
- ^ Milne, Theorem 9.13.
- ^ Milne, Lemma 9.6.
- .
- ^ Hartshorne 1977, Ch I, § 4, just before Theorem 4.7.A
- ^ Hartshorne 1977, Ch I, Theorem 4.7.A
- ^ Milne, Theorem 9.27.
- ^ Milne, Proposition 9.16.
- ^ Hartshorne 1977, Ch. II, Theorem 8.6. A