Complete splitting and unramifiedness in composita and Galois closures #
Let M / K be a finite Galois extension of number fields with group G, let E be an
intermediate field, and let ๐ญ be a prime of ๐ K. Say that ๐ญ splits completely in E
when it has [E : K] primes above it in ๐ E. This file proves
๐ญsplits completely inEexactly when every decomposition groupD(Q) โค G, forQa prime of๐ Mabove๐ญ, fixesEpointwise;๐ญis unramified inEexactly when every inertia groupI(Q)fixesEpointwise.
Both conditions are about the fixing subgroup of E, and the Galois correspondence turns a join
of fields into a meet of fixing subgroups. So both properties pass to composita: ๐ญ splits
completely (or is unramified) in Eโ โ Eโ exactly when it does in Eโ and in Eโ, and likewise
for an arbitrary join. Conjugating E by ฯ โ G conjugates its fixing subgroup, while the
decomposition and inertia groups over ๐ญ are permuted among themselves by the same conjugation;
so neither property changes. As the normal closure of E in M is the join of the conjugates of
E, the prime ๐ญ splits completely (or is unramified) in E exactly when it does in the normal
closure. When M is itself the normal closure of E, this compares E with M directly.
The decomposition-group criterion rests on the local degree formula
e(Q โฉ E / ๐ญ) f(Q โฉ E / ๐ญ) = [D(Q) : D(Q) โฉ Gal(M/E)], together with the non-Galois count
criterion "[E : K] primes above ๐ญ exactly when e = f = 1 at each". The inertia-group
criterion rests likewise on e(Q โฉ E / ๐ญ) = [I(Q) : I(Q) โฉ Gal(M/E)].
Main results #
Ideal.ncard_primesOver_eq_finrank_iff_forall_stabilizer_le:๐ญsplits completely inEiff every decomposition group above๐ญlies inGal(M/E).Ideal.ncard_primesOver_eq_finrank_iff_forall_stabilizer_eq_bot:๐ญsplits completely inMiff every decomposition group above๐ญis trivial.Ideal.ncard_primesOver_sup_eq_finrank_iff,Ideal.ncard_primesOver_iSup_eq_finrank_iff: complete splitting in a compositum.Ideal.ncard_primesOver_map_eq_finrank_iff: complete splitting in a conjugate field.Ideal.ncard_primesOver_normalClosure_eq_finrank_iff:๐ญsplits completely inEiff it splits completely in the normal closure ofE.Ideal.ncard_primesOver_eq_finrank_iff_of_normalClosure_eq_top: the same, whenMis the normal closure ofE.Ideal.isUnramifiedIn_iff_forall_inertia_le:๐ญis unramified inEiff every inertia group above๐ญlies inGal(M/E).Ideal.isUnramifiedIn_sup_iff,Ideal.isUnramifiedIn_iSup_iff,Ideal.isUnramifiedIn_map_iff,Ideal.isUnramifiedIn_normalClosure_iff: the same four consequences for unramifiedness.
References #
- [J. Neukirch, Algebraic Number Theory][Neukirch1992], Chapter I, ยง9, Exercise 4.
Complete splitting #
Complete splitting through decomposition groups. For M / K Galois and E an
intermediate field, a prime p of ๐ K has [E : K] primes above it in ๐ E exactly when the
decomposition group of every prime of ๐ M above p fixes E pointwise.
Complete splitting in a Galois extension. For M / K Galois, a prime p of ๐ K has
[M : K] primes above it exactly when the decomposition group of every prime above it is
trivial.
Complete splitting in a compositum. A prime p of ๐ K splits completely in Eโ โ Eโ
exactly when it splits completely in Eโ and in Eโ.
Complete splitting in an arbitrary compositum. A prime p of ๐ K splits completely in
โจ i, E i exactly when it splits completely in every E i.
Complete splitting in a conjugate field. A prime p of ๐ K splits completely in
ฯ E exactly when it splits completely in E.
Complete splitting in the normal closure. A prime p of ๐ K splits completely in an
intermediate field E of M / K exactly when it splits completely in the normal closure of E
in M.
Complete splitting in a field and in its Galois closure. If M is the normal closure of
the intermediate field E, a prime p of ๐ K splits completely in E exactly when it splits
completely in M.
Unramifiedness #
Unramifiedness through inertia groups. For M / K Galois and E an intermediate field,
an ideal p of ๐ K is unramified in E exactly when the inertia group of every prime of ๐ M
above p fixes E pointwise.
Unramifiedness in a compositum. An ideal p of ๐ K is unramified in Eโ โ Eโ exactly
when it is unramified in Eโ and in Eโ.
Unramifiedness in an arbitrary compositum. An ideal p of ๐ K is unramified in
โจ i, E i exactly when it is unramified in every E i.
Unramifiedness in a conjugate field. An ideal p of ๐ K is unramified in ฯ E exactly
when it is unramified in E.
Unramifiedness in the normal closure. An ideal p of ๐ K is unramified in an
intermediate field E of M / K exactly when it is unramified in the normal closure of E
in M.