The genus of y ^ 2 = f(x) #
Let k be a field of characteristic other than two and F / k(x) an extension generated by an
element y with y ^ 2 = f(x), where f ∈ k[X] is squarefree of degree m ≥ 1; then F / k(x)
is finite and separable.
This file computes the invariants of F / k from the ramification of F / k(x):
F / k(x)has degree two, andkis the exact field of constants ofF;- the places of
k(x)that ramify inFare the zeros off, together with the place at infinity whenmis odd; - the different of
F / k(x)has degreem + (m mod 2), so the Hurwitz genus formula2g - 2 = -2 · 2 + deg Diff(F / k(x))givesg = ⌊(m - 1) / 2⌋; - for
m ≥ 5,F / kis a hyperelliptic function field.
For odd m, the place at infinity is totally ramified with a unique rational place above it
in every characteristic, without squarefreeness; these models also have exact constants.
The local input is the ramification of the radical extension y ^ 2 = f at a place P of k(x)
(TauCeti.Place.ramificationIdx_eq_of_pow_eq_of_prime and
TauCeti.Place.differentExponent_eq_of_pow_eq_of_prime): the places above P are ramified, with
different exponent one, exactly when ord_P f is odd. Since f is a squarefree polynomial,
ord_P f is 0 or 1 at a finite place, and -m at infinity. To add up the different without
counting the places above each ramified P, compare it with the conorm of the divisor B of
k(x) consisting of the places where ord_P f is odd: every place above B has ramification
index two, so Con B = 2 · Diff(F / k(x)), and the conorm multiplies degrees by [F : k(x)] = 2.
Hence deg Diff(F / k(x)) = deg B, and deg B = m + (m mod 2) because the zeros of f have
total degree m by the product formula.
Exactness of the constants comes from the ramification as well: a finite separable extension with
a totally ramified place acquires no new constants
(TauCeti.isIntegrallyClosedIn_of_isTotallyRamified).
Main results #
TauCeti.finrank_ratFunc_eq_two_of_sq_eqandTauCeti.isIntegrallyClosedIn_of_sq_eq: the degree ofF / k(x)and exactness of the constant fieldk.TauCeti.finiteDimensional_and_isSeparable_of_sq_eq: squarefree nonconstant models are finite and separable away from characteristic two.TauCeti.finrank_ratFunc_eq_two_of_sq_eq_of_odd: odd-degree models have degree two.TauCeti.isIntegrallyClosedIn_of_sq_eq_of_odd: odd-degree models have exact constants without squarefreeness or characteristic assumptions.TauCeti.Place.isTotallyRamified_of_sq_eq_of_oddandTauCeti.Place.existsUnique_restrict_eq_infty_of_sq_eq_of_odd: total ramification at infinity and the unique rational place above it for odd-degree models, without squarefreeness or characteristic assumptions.TauCeti.Place.one_lt_ramificationIdx_iff_of_sq_eq: the ramified places.TauCeti.degree_different_ratFunc_of_sq_eq:deg Diff(F / k(x)) = m + (m mod 2).TauCeti.genus_eq_of_sq_eq: the genus ofy ^ 2 = f(x)is⌊(m - 1) / 2⌋.TauCeti.isHyperellipticFunctionField_of_sq_eq: form ≥ 5the field is hyperelliptic.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Example 3.7.6 and Proposition 6.2.3.
The branch divisor on k(x) #
The degree of y ^ 2 = f #
y ^ 2 = f has degree two over k(x) for a squarefree nonconstant polynomial f. No
hypothesis on the characteristic is needed.
A squarefree nonconstant polynomial model generates a finite separable extension of k(x)
away from characteristic two.
An odd-degree polynomial model has degree two over k(x), in every characteristic
and without squarefreeness.
Ramification and the different of y ^ 2 = f #
In a polynomial model of odd degree, every place above infinity is totally ramified. This also holds in characteristic two, without squarefreeness.
An odd-degree polynomial model has a unique place above infinity; that place has degree one. No squarefreeness, separability, or characteristic assumption is needed.
The constants of an odd-degree polynomial model are exactly k.
No squarefreeness, separability, or characteristic assumption is needed.
The ramified places of y ^ 2 = f (Stichtenoth, Proposition 6.2.3): a place of F is
ramified over k(x) exactly when it lies over a zero of f, or over the place at infinity with
deg f odd. Its ramification index is then two, by
TauCeti.Place.ramificationIdx_eq_of_pow_eq_of_prime.
Away from the branch locus, a place over a finite place q of k(x) not dividing f is
unramified over k(x): it lies neither over infinity nor over a zero of f.
The degree of the different of y ^ 2 = f: for a squarefree polynomial f of degree
m ≥ 1, the different of F / k(x) has degree m + (m mod 2), the number of branch points
counted with their degrees.
The constants and the genus of y ^ 2 = f #
The constant field of y ^ 2 = f is exact: for a squarefree nonconstant polynomial f,
k is algebraically closed in F = k(x, y).
The genus of y ^ 2 = f(x) (Stichtenoth, Example 3.7.6 and Proposition 6.2.3): away from
characteristic two, if F = k(x, y) with y ^ 2 = f(x) for a squarefree polynomial f of degree
m ≥ 1, then F / k has genus ⌊(m - 1) / 2⌋: (m - 1) / 2 for odd m and (m - 2) / 2 for
even m.
y ^ 2 = f(x) is hyperelliptic when deg f ≥ 5 (Stichtenoth, Proposition 6.2.3): away
from characteristic two, if F = k(x, y) with y ^ 2 = f(x) for a squarefree polynomial f of
degree at least five, then F / k is a hyperelliptic function field.