Constant field extensions of the rational function field #
For a field extension k' / k, the rational function field k'(X) is the constant field
extension k(X) · k' of k(X), for any k(X)-algebra structure on k'(X) sending X to X.
Such a structure maps a polynomial to the polynomial with the same coefficients read in k'.
When k' / k is finite, the place at infinity of k'(X) is the only place over the place at
infinity of k(X).
The finite places are described by factorization. If q ∈ k[X] is irreducible, the places of
k'(X) over the finite place of q are the finite places of the irreducible factors q' of q
in k'[X], and the ramification index of the place of q' is the multiplicity of q' in q.
Neither separability of k' / k nor of q is assumed. When q is inseparable this
multiplicity can exceed one: over a field of characteristic p, X ^ p - a becomes
(X - a ^ (1 / p)) ^ p once a p-th root of a is adjoined. At a rational finite place, the
place of X - c for c ∈ k, it cannot: the place of X - c in k'(X) is the only place above,
with ramification index one and relative degree [k' : k], and the conorm of the place is that
single place.
Main results #
TauCeti.constantCompositum_eq_top_of_X_mem: a compositumF · k'insidek'(X)containingXis all ofk'(X).TauCeti.constantCompositum_ratFunc_eq_top:k'(X)is the compositum ofk(X)andk'.TauCeti.constantCompositum_adjoin_simple_ratFunc_eq_top: ifzgeneratesEoverk'(X), thenEis the compositum ofk(X)(z)andk'.TauCeti.Place.restrict_ratFunc_eq_infty_iff: a place ofk'(X)lies over the place at infinity ofk(X)exactly when it is the place at infinity ofk'(X).TauCeti.Place.restrict_adicOfIrreducible_eq_iff: the finite place ofq'lies over the finite place ofqexactly whenq'dividesqink'[X].TauCeti.Place.restrict_eq_adicOfIrreducible_iff: the places over the finite place ofqare the finite places of the irreducible factors ofqink'[X].TauCeti.Place.ramificationIdx_adicOfIrreducible_eq_multiplicity: the ramification index of the place of such a factor is its multiplicity inq.TauCeti.Place.restrict_eq_adicOfIrreducible_X_sub_C_iff,TauCeti.Place.ramificationIdx_adicOfIrreducible_X_sub_C_algebraMapandTauCeti.Place.conorm_ofPoint_adicOfIrreducible_X_sub_C: forc ∈ k, the place ofX - cink'(X)is the only place over the place ofX - cink(X), withe = 1andCon P = Q.TauCeti.Place.relativeDegree_adicOfIrreducible_X_sub_C_algebraMap: at that placef = [k' : k], since the residue extension isk' / k.TauCeti.Place.isUnramified_adicOfIrreducible_X_sub_C_iff: the place ofX - cink(X)is unramified ink'(X)exactly whenk' / kis separable.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Section 3.6.
A compositum with k' containing X is all of k'(X): for any field F inside k'(X),
if X lies in F · k', then F · k' = k'(X).
k'(X) is the compositum of k(X) with the constants k', for any k(X)-algebra
structure on k'(X) sending X to X.
A simple extension descended to k(X) is a model. If z generates E over k'(X), then
the compositum of k(X)(z) with the constants k' is all of E, for any k(X)-algebra
structure on k'(X) sending X to X.
The place at infinity of k'(X) is the only place over the place at infinity of k(X),
for a finite extension k' / k and any k(X)-algebra structure on k'(X) sending X to X.
The places over a finite place #
A finite place of k'(X) lies over a finite place of k(X): X is regular at it.
The finite place of q' lies over the finite place of q exactly when q' divides q in
k'[X], for irreducible q ∈ k[X] and q' ∈ k'[X].
The places over the finite place of q are the finite places of the irreducible factors
of q in k'[X].
The ramification index at an irreducible factor is its multiplicity: if the irreducible
q' ∈ k'[X] divides the irreducible q ∈ k[X], then e at the finite place of q' over the
finite place of q is the multiplicity of q' in q.
Rational finite places #
The place of X - c in k'(X), for c ∈ k, lies over the place of X - c in k(X).
Over a rational finite place there is only one place: for c ∈ k, the place of X - c
in k'(X) is the only place over the place of X - c in k(X).
A rational finite place has ramification index one in k'(X) / k(X): for c ∈ k,
e = 1 at the place of X - c in k'(X).
A rational finite place has relative degree [k' : k] in k'(X) / k(X): for c ∈ k,
the place of X - c in k'(X) and the place below it are both rational, so the residue extension
is k' / k.
A rational finite place of k(X) is unramified in k'(X) exactly when k' / k is
separable, when k(X) is embedded in k'(X) by X ↦ X. For c ∈ k, the only place above the
place of X - c is the place of X - c in k'(X), with ramification index one; both places are
rational, so the residue extension is k' / k.
The conorm of a rational finite place is a single place: for c ∈ k, the conorm of the
place of X - c in k(X) is the place of X - c in k'(X).