Ramification in a radical extension y ^ n = u #
Let F' / k' be an extension of the field extension F / k, and let y ∈ F' satisfy
y ^ n = u for some u ∈ F. At a place P' of F' over the place P of F, taking orders
in y ^ n = u gives
n · ord_{P'} y = e(P' ∣ P) · ord_P u.
Writing r_P = gcd(n, ord_P u), the two quotients n / r_P and ord_P u / r_P are coprime, so
n / r_P divides the ramification index. This is the lower half of the ramification data of a
Kummer extension (Stichtenoth, Proposition 3.7.3(b), where e(P' ∣ P) = n / r_P), and it holds
in every characteristic and without any root of unity in the constants. The upper half needs
char k ∤ n: it is the statement that adjoining an r_P-th root of a unit of 𝒪_P is
unramified.
When n is coprime to ord_P u the lower bound is already the whole degree if [F' : F] = n.
If y generates F' over F, this degree equality follows from
Valuation.finrank_eq_of_pow_eq_of_gcd_ord_eq_one. Then n ∣ e(P' ∣ P) forces
e(P' ∣ P) = n: the place P is totally ramified in F', and ord_{P'} y = ord_P u.
This covers, for instance, the places of k(x) at the simple zeros of a squarefree f in
y ^ 2 = f(x), and the place at infinity when f has odd degree.
Main results #
TauCeti.Place.natCast_mul_ord_eq_ramificationIdx_mul_ord_of_pow_eq:n · ord_{P'} y = e(P' ∣ P) · ord_P u.TauCeti.Place.div_gcd_ord_dvd_ramificationIdx_of_pow_eq:n / gcd(n, ord_P u)dividese(P' ∣ P).TauCeti.Place.isTotallyRamified_of_pow_eq_of_gcd_ord_eq_oneandTauCeti.Place.ramificationIdx_eq_of_pow_eq_of_gcd_ord_eq_one: if[F' : F] = nandgcd(n, ord_P u) = 1, thenP'is totally ramified overFwithe(P' ∣ P) = n.TauCeti.Place.ord_eq_of_pow_eq_of_gcd_ord_eq_one: in that caseord_{P'} y = ord_P u.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Proposition 3.7.3.
Orders in a radical extension: if y ^ n = u with u ∈ F, then at every place P' of
F' the orders of y and u are related by n · ord_{P'} y = e(P' ∣ P) · ord_P u, where P is
the place of F below P'.
The ramification lower bound of a radical extension (Stichtenoth, Proposition 3.7.3(b)):
if y ^ n = u with u ∈ F and n ≠ 0, then n / gcd(n, ord_P u) divides the ramification index
e(P' ∣ P). No hypothesis on the characteristic or on roots of unity is needed; for u = 0 the
order is the junk value 0 and the statement is trivial.
If F' = F(y) with y ^ n = u and n coprime to ord_P u, then some generator z of
F' / F has z ^ n of order one at P.
Total ramification in a radical extension (Stichtenoth, Proposition 3.7.3(b)): if
[F' : F] = n with y ^ n = u, n ≠ 0, and n is coprime to the order of u at the place P
of F below P', then e(P' ∣ P) = n.
Total ramification in a radical extension (Stichtenoth, Proposition 3.7.3(b)): if
[F' : F] = n with y ^ n = u, n ≠ 0, and n is coprime to the order of u at the place P
of F below P', then P' is totally ramified over F. With
TauCeti.Place.setOf_restrict_eq_eq_singleton_of_isTotallyRamified this says that P' is the
only place of F' over P, with relative degree 1.
The order of the radical at a totally ramified place: if [F' : F] = n with y ^ n = u,
n ≠ 0, and n is coprime to the order of u at the place P of F below P', then
ord_{P'} y = ord_P u. In particular y is a prime element at P' when u is one at P.