The existence theorem

Section 7.4 The existence theorem

Reference.

[37] VII.6, VII.9, [38] VI.4, VI.6.

With the reciprocity isomorphism in hand, we now prove the existence theorem in its idelic formulation (see Theorem 6.4.4). As a reminder, deducing the ideal-theoretic existence theorem (Theorem 2.2.8) will also require local-global compatibility (Proposition 7.5.6), as described in Remark 6.4.10.

🔗

As in the proof of the local existence theorem (Theorem 4.3.18), having access to the reciprocity law, even in abstract form, reduces the task of proving the existence theorem to the “topological” assertion that every open subgroup of

C_{L}

of finite index contains a norm subgroup. For this, we can essentially rerun the Kummer-theoretic argument from the local case. Since this argument is agnostic as to where the reciprocity map came from, it also applies in the approach that does not use abstract class field theory (Section 7.6).

🔗

After proving the existence theorem, we give the closely related algebraic proof of the Second Inequality (Theorem 7.2.9).

🔗

Subsection A base case for the existence theorem

As in the proof of the local existence theorem (Theorem 4.3.18), the key to the proof of Theorem 7.4.8 is showing that for any given number field

K,

we can find finite extensions

L / K

for which the groups

{Norm}_{L / K} C_{L}

can be made arbitrarily small. In preparation for an inductive proof, we establish a key base case using Kummer theory.

🔗

Lemma 7.4.1.

Let

K

be a number field containing a primitive

p

-th root of unity for some prime

p .

Let

U

be an open subgroup of

C_{K}

of index

p .

Then for some finite set

S

of places of

K

containing the infinite places and all places above

p,

I_{K} = K^{*} I_{K, S}

and the preimage of

U

I_{K, S}

contains

🔗

W_{S} := \prod_{v \in S} (K_{v}^{*})^{p} \times \prod_{v \notin S} o_{K_{v}}^{*} .

🔗

Proof.

Let

J

be the preimage of

U

under the projection

I_{K} \to C_{K},

so that

J

is open in

I_{K}

of index

p .

Then

J

contains a subgroup of the form

V := \prod_{v \in S} {1} \times \prod_{v \notin S} o_{K_{v}}^{*}

for some finite set

S

of places of

K

containing the infinite places, which by Corollary 6.2.10 we may choose large enough so that

I_{K} = K^{*} I_{K, S} .

The group

J

must also contain

I_{K}^{p},

and hence

W_{S} .

We continue with a lemma that allows to detect whether certain elements of a number field are

p

-th powers based on whether this happens locally. This amounts to a carefully chosen special case of the Grunwald-Wang theorem (Remark 7.2.12).

🔗

Lemma 7.4.2.

With notation as in Lemma 7.4.1,

🔗

W_{S} \cap K^{*} = (o_{K, S}^{*})^{p} .

🔗

Proof.

It is clear that

(o_{K, S}^{*})^{p} \subseteq W_{S} \cap K^{*} .

To prove the reverse inclusion, note that for any

y \in W_{S} \cap K^{*},

if we set

L = K (y^{1 / p}),

then every place

v \in S

is split in

L

and every place

v \notin S

is unramified in

L,

yielding

{Norm}_{L / K} I_{L, S} = I_{K, S} .

Since

I_{K} = K^{*} I_{K, S},

this implies

{Norm}_{L / K} C_{L} = C_{K} .

By the First Inequality (Theorem 7.1.2), this implies

L = K

and so

y \in (K^{*})^{p} .

This will in turn enable us to compute the norm group for a certain compositum of Kummer extensions.

🔗

Lemma 7.4.3.

With notation as in Lemma 7.4.1, put

s = # S

and

🔗

L = K (u^{1 / p} : u \in o_{K, S}^{*}) .

Then

[L : K] = p^{s}

and

🔗

{Norm}_{L / K} C_{L} = K^{*} W_{S} / K^{*} \subset C_{K} .

🔗

Proof.

By Corollary 6.2.11 and the assumption that

K

contains a primitive

p

-th root of unity, the group

o_{K, S}^{*} / (o_{K, S}^{*})^{p}

is finite of order

p^{s},

yielding

[L : K] = p^{s} .

By local reciprocity (Lemma 4.3.13), we have

K^{*} W_{S} / K^{*} \subseteq {Norm}_{L / K} C_{L};

to prove equality, it will suffice to show that these subgroups have the same finite index in

C_{K} .

Consider now the exact sequence

1 \to \frac{o_{K, S}^{*}}{o_{K, S}^{*} \cap W_{S}} \to \frac{I_{K, S}}{W_{S}} \to \frac{C_{K}}{K^{*} W_{S} / K^{*}} \to 1 .

By Lemma 7.4.2, the group on the left has order

[o_{K, S}^{*} : (o_{K, S}^{*})^{p}] = p^{s} .

By Lemma 7.4.4, the group in the middle has order

p^{2 s} .

We thus have

[C_{K} : K^{*} W_{S} / K^{*}] = p^{s} = [L : K] = [C_{K} : {Norm}_{L / K} C_{L}]

where the last equality follows from the global reciprocity isomorphism (Theorem 6.4.3).

Here is the local calculation used in the proof of Lemma 7.4.3.

🔗

Lemma 7.4.4.

For

K

a number field,

v

a place of

K,

and

p

a prime such that

ζ_{p} \in K,

🔗

[K_{v}^{*} : (K_{v}^{*})^{p}] = \frac{p^{2}}{| p |_{v}} .

🔗

Proof.

We separate cases as follows.

If $v$ is a real place, then $p = 2,$ $\frac{p^{2}}{| p |_{v}} = 2,$ and
$K_{v}^{*} / (K_{v}^{*})^{p} = R^{*} / (R^{*})^{2} = R^{*} / R^{+} ≅ Z / 2 Z .$
If $v$ is a complex place, then $\frac{p^{2}}{| p |_{v}} = 1$ according to our conventions (Definition 6.1.11), and $K_{v}^{*} / (K_{v}^{*})^{p}$ is trivial because $C^{*}$ is $p$ -divisible.
If $v$ is a finite place not lying above $p,$ then $\frac{p^{2}}{| p |_{v}} = p^{2}$ and $K_{v}^{*} / (K_{v}^{*})^{p}$ is generated by $ζ_{p}$ and a uniformizer of $K_{v} .$
If $v$ is a finite place above $p,$ then $| p |_{v} = p^{- n}$ for some positive integer $n,$ so $\frac{p^{2}}{| p |_{v}} = p^{n + 2} .$ Since $K_{v}^{*} ≅ o_{K_{v}}^{*} \times Z,$ it suffices to check that $[o_{K_{v}}^{*} : (o_{K_{v}}^{*})^{p}] = p^{n + 1} .$ For this, see Exercise 1.

We finally put everything together to get a key special case of the existence theorem.

🔗

Lemma 7.4.5.

Let

K

be a number field containing a primitive

p

-th root of unity for some prime

p .

Let

U

be an open subgroup of

C_{K}

of index

p .

Then there exists a finite extension

L

K

such that

{Norm}_{L / K} C_{L} \subseteq U .

🔗

Proof.

This now follows from Lemma 7.4.1 and Lemma 7.4.3. which by Corollary 6.2.10 we may choose large enough so that

K^{*} I_{K, S} = I_{K} .

🔗

Subsection Proof of the existence theorem

Building on the base case offered by Lemma 7.4.5, we now finish the proof of the existence theorem.

🔗

Lemma 7.4.6.

Let

K

be a number field. Let

U

be an open subgroup of

C_{K}

of some prime index

p .

Then there exists a finite extension

L

K

such that

{Norm}_{L / K} C_{L} \subseteq U .

🔗

Proof.

Take

K^{'} = K (ζ_{p}) .

Let

U^{'}

be the inverse image of

U

C_{K^{'}} .

By Theorem 6.4.3

[C_{K} : {Norm}_{K^{'} / K} C_{K^{'}}] = [K^{'} : K]

is coprime to

p;

consequently,

[C_{K^{'}} : U^{'}] = p .

By Lemma 7.4.5, there exists a finite extension

L / K^{'}

such that

{Norm}_{L / K^{'}} C_{L} \subseteq U^{'};

then

{Norm}_{L / K} C_{L} \subseteq {Norm}_{K^{'} / K} U^{'} \subseteq U .

Lemma 7.4.7.

Let

K

be a number field. Let

U

be an open subgroup of

C_{K}

of finite index. Then there exists a finite extension

L

K

such that

{Norm}_{L / K} C_{L} \subseteq U .

🔗

Proof.

We proceed by induction on the index

[C_{K} : U],

with Lemma 7.4.6 as the base case. Otherwise, choose an intermediate subgroup

V

between

U

and

C_{K} .

By the induction hypothesis,

V

contains

N := {Norm}_{L / K} C_{L}

for some finite extension

L

K .

Then

[N : (U \cap N)] = [U N : U] \leq [V : U] .

Let

W

be the subgroup of

C_{L}

consisting of those

x

whose norms lie in

U .

Then

[C_{L} : W] \leq [N : U \cap N] \leq [V : U],

so by the induction hypothesis

W

contains

{Norm}_{M / L} C_{M}

for some finite extension

M / L .

Thus

U

contains

{Norm}_{M / K} C_{M},

as desired.

Theorem 7.4.8. Adelic existence theorem.

For

K

a number field, the finite abelian extensions

L / K

are in bijection with the open subgroups of

C_{K}

of finite index via the map

L \mapsto {Norm}_{L / K} C_{L} .

🔗

Proof.

For any finite abelian extension

L / K,

{Norm}_{L / K} C_{L}

is a subgroup of

C_{K}

which is open (by Lemma 6.4.1) of index

[L : K]

(by Theorem 6.4.3). Moreover, by Corollary 5.3.13, the correspondence

L \mapsto {Norm}_{L / K} C_{L}

is injective.

Conversely, let

U

be an open subgroup of

C_{K}

of finite index. By Lemma 7.4.7, there exists a finite extension

L_{1} / K

such that

{Norm}_{L_{1} / K} C_{L_{1}} \subseteq U .

By the adelic norm limitation theorem (Theorem 6.4.5), we also have

{Norm}_{L_{1} / K} C_{L_{1}} = {Norm}_{L_{2} / K} C_{L_{2}} \subseteq U

for

L_{2} / K

the maximal abelian subextension of

L_{1} / K .

By Theorem 6.4.3 again, we have an isomorphism

Gal (L_{2} / K) ≅ C_{K} / {Norm}_{L_{2} / K} C_{L_{2}},

via which the subgroup

U / {Norm}_{L_{2} / K} C_{L_{2}}

corresponds to a subgroup

H

Gal (L_{2} / K) .

Taking

L

to be the fixed field of

H,

we deduce that

{Norm}_{L / K} C_{L} = U

as desired.

Remark 7.4.9.

As with the proof of the local existence theorem, the proof of Theorem 7.4.8 is constructive in principle but not in practice: it involves constructing some extension much larger than the desired abelian extension, then invoking the norm limitation theorem to get down to an abelian extension. We remind the reader that there is no easy fix known for this (Remark 2.2.10).

🔗

Subsection An algebraic approach to the Second Inequality

Drawing inspiration from the calculation of norm groups given in Lemma 7.4.3, we now explain how to use similar ideas to give an algebraic proof of the Second Inequality. Again, the key case is where

L / K

is a cyclic extension of number fields of prime degree

p

and

ζ_{p} \in K .

To modify the calculation from Lemma 7.4.3 to compute the norm group of a single Kummer extension, we use a second set of places.

🔗

Lemma 7.4.10.

Let

K

be a number field containing

ζ_{p}

for some prime

p .

Let

L / K

be a cyclic extension of number fields of degree

p .

For some positive integer

s,

we can then choose the following.

🔗

A finite set $S$ of $s$ places of $K$ containing all infinite places, all places that ramify in $L,$ and all places above $p,$ for which $I_{K} = I_{K, S} K^{*} .$
A second set $T$ of $s - 1$ places of $K$ disjoint from $S,$ such that $o_{K, S}^{*} \to \prod_{v \in T} K_{v}^{*} / (K_{v}^{*})^{p}$ is surjective with kernel $Δ$ and $L = K (Δ^{1 / p}) .$

🔗

Proof.

By Kummer theory (Theorem 1.2.6), we can choose a finite set

S

of places of

K

containing all infinite places, all places that ramify in

L,

and all places above

p

so that

L = K (Δ^{1 / p})

for

Δ = o_{K, S}^{*} \cap (L^{*})^{p} .

This remains true after enlarging

S,

so by Corollary 6.2.10 we can further ensure that

I_{K} = I_{K, S} K^{*} .

Put

N := K ((o_{K, S}^{*})^{1 / p}) .

By Kummer theory again,

Gal (N / K) ≅ Hom (o_{K, S}^{*} / (o_{K, S}^{*})^{p}, Z / p Z) .

By Corollary 6.2.11,

o_{K, S}^{*} / (o_{K, S}^{*})^{p} ≅ (Z / p Z)^{s} .

Choose generators

g_{1}, \dots, g_{s - 1}

Gal (N / L);

these correspond in

Hom (o_{K, S}^{*} / (o_{K, S}^{*})^{p}, Z / p Z)

to a set of homomorphisms whose common kernel is precisely

Δ / (o_{K, S}^{*})^{p} .

We thus need to find, for each

g_{i},

a place

v_{i}

such that the kernel of

g_{i}

is the same as the kernel of

o_{K, S}^{*} \to K_{v_{i}}^{*} / (K_{v_{i}}^{*})^{p};

we can then take

T = {v_{1}, \dots, v_{s - 1}} .

Let

N_{i}

be the fixed field of

g_{i};

by Corollary 7.1.15 (which we deduced from the First Inequality) or Proposition 7.2.3 (which was part of the analytic proof of the Second Inequality), there are infinitely many primes of

N_{i}

that do not split in

N .

So we can choose a place

w_{i}

of each

N_{i}

such that their restrictions

v_{i}

K

are distinct, not contained in

S,

and don’t divide

p .

We claim

N_{i}

is the maximal subextension of

N / K

in which

v_{i}

splits completely (i.e., the decomposition field of

v_{i}

). On one hand,

v_{i}

does not split completely in

N,

so the decomposition field is no larger than

N_{i} .

On the other hand, the decomposition field is the fixed field of the decomposition group, which has exponent

p

and is cyclic (since

v_{i}

does not ramify in

N

). Thus it must have index

p

N,

so must be

N_{i}

itself.

Thus

L = ⋂ N_{i}

is the maximal subextension of

N

in which all of the

v_{i}

split completely. We conclude that for

x \in o_{K, S}^{*},

x

belongs to

Δ

iff

K_{v_{i}} (x^{1 / p}) = K_{v_{i}}

for all

i,

which occurs iff

x \in K_{v_{i}}^{p} .

That is,

Δ

is precisely the kernel of the map

o_{K, S}^{*} \to \prod_{i} K_{v_{i}}^{*} / (K_{v_{i}}^{*})^{p} .

This proves the claim.

We have the following modified version of Lemma 7.4.2.

🔗

Lemma 7.4.11.

With notation as in Lemma 7.4.10, put

🔗

W_{S, T} := \prod_{v \in S} (K_{v}^{*})^{p} \times \prod_{v \in T} K_{v}^{*} \times \prod_{v \notin S \cup T} o_{K_{v}}^{*} .

Then

🔗

W_{S, T} \cap K^{*} = (o_{K, S \cup T}^{*})^{p} .

🔗

Proof.

It is again clear that

(o_{K, S \cup T}^{*})^{p} \subseteq W_{S, T} \cap K^{*} .

To prove the reverse inclusion, it will again suffice to prove that

y \in W_{S, T} \cap K^{*},

if we set

L = K (y^{1 / p}),

then

{Norm}_{L / K} C_{L} = C_{K};

namely, Theorem 7.1.2 will then imply

L = K

and so

y \in (K^{*})^{p} .

Since

o_{K, S}^{*} \to \prod_{v \in T} o_{K_{v}}^{*} / (o_{K_{v}}^{*})^{p}

is surjective, any element of

I_{K, S \cup T}

can be written as the product of an element of

o_{K, S}^{*}

with an element of

I_{K, S \cup T}

which is a

p

-th power at each place of

T .

In particular, by Lemma 4.3.13 such an element is a norm from

L

at each place of

T;

we can now reprise the proof of Lemma 7.4.2, skipping over the places in

T,

to deduce that we have a norm from

L .

Lemma 7.4.12.

With notation as in Lemma 7.4.10 and Lemma 7.4.11,

K^{*} W_{S, T} / K^{*}

is contained in

{Norm}_{L / K} C_{L}

and has index

p

C_{K} .

Consequently, the Second Inequality holds for

L / K .

🔗

Proof.

By Lemma 7.4.11, we have an exact sequence

1 \to \frac{o_{K, S \cup T}^{*}}{(o_{K, S \cup T}^{*})^{p}} \to \frac{I_{K, S \cup T}}{W_{S, T}} \to \frac{C_{K}}{K^{*} W_{S, T} / K^{*}} \to 1,

with which we may compute as in Lemma 7.4.3: the left group has order

p^{# (S \cup T)} = p^{2 s - 1}

by Corollary 6.2.11 while the middle group has order

p^{2 s}

by Lemma 7.4.4, so

[C_{K} : K^{*} W_{S, T} / K^{*}] = p .

Meanwhile, we can check by local reciprocity that

{Norm}_{L / K} I_{L, S} = I_{K, S}

(compare the proof of Lemma 7.4.11).

For $v \in S,$ elements of $(K_{v}^{*})^{p}$ are norms from any abelian extension of $K_{v}$ of exponent $p$ (by Lemma 4.3.13).
For $v \in T,$ $v$ splits in $L$ and so $L_{w} = K_{v} .$
For $v \notin S \cup T,$ $v$ is unramified in $L$ and so ${Norm}_{L_{w} / K_{v}} o_{L_{w}}^{*} = o_{K_{v}}^{*} .$

Hence

K^{*} W_{S, T} \subseteq {Norm}_{L / K} C_{L},

completing the proof.

Lemma 7.4.13.

Let

L / K

be a cyclic extension of number fields of prime degree

p

and let

K^{'} = K (ζ_{p}), L^{'} = L (ζ_{p}) .

Then the map

🔗

H_{T}^{0} (Gal (L / K), C_{L}) \to H_{T}^{0} (Gal (L^{'} / K^{'}), C_{L^{'}})

induced by the inclusion

C_{L} \to C_{L^{'}}

is injective.

🔗

Proof.

For

x \in C_{K},

{Norm}_{L / K} (x) = x^{p};

this implies that both groups in question are killed by

p .

In particular, multiplication by

d = [K^{'} : K],

which divides

p - 1,

is an isomorphism on these groups.

Suppose

x \in C_{K}

maps to the identity in

H_{T}^{0} (Gal (L^{'} / K^{'}), C_{L^{'}}) .

We can then choose a representative of the class of

x

H_{T}^{0} (Gal (L / K), C_{L})

of the form

y^{d};

then

y

also maps to the identity in

H_{T}^{0} (Gal (L^{'} / K^{'}), C_{L^{'}}) .

That is,

y = {Norm}_{L^{'} / K^{'}} (z^{'})

for some

z^{'} \in C_{L^{'}},

and

y^{d} = {Norm}_{K^{'} / K} (y) = {Norm}_{L^{'} / K} (z^{'}) \in {Norm}_{L / K} C_{L} .

Thus

x \in {Norm}_{L / K} C_{L},

as needed.

Theorem 7.4.14. Second Inequality (algebraic proof).

Let

L / K

be a Galois extension of number fields with Galois group

G .

Then:

🔗

the group $H^{1} (G, C_{L})$ is trivial;
the group $H^{2} (G, C_{L})$ is finite of order at most $[L : K] .$

🔗

Proof.

As in the proof of Theorem 7.2.9, we use an induction argument to reduce to proving that for

L / K

cyclic,

H_{T}^{0} (Gal (L / K), C_{L}) \leq [L : K] .

In fact, the same induction (considering

H^{2}

in place of

H_{T}^{0}

) allows us to further reduce to the case where

[L : K] = p

is prime.

Let

K^{'} = K (ζ_{p})

and

L^{'} = L (ζ_{p});

then

K^{'}

and

L

are linearly disjoint over

K

(since their degrees are coprime), so

[L^{'} : K^{'}] = [L : K] = p

and the Galois groups of

L / K

and

L^{'} / K^{'}

are canonically isomorphic. By Lemma 7.4.12 and Lemma 7.4.13,

# H_{T}^{0} (Gal (L / K), C_{L}) \leq # H_{T}^{0} (Gal (L^{'} / K^{'}), C_{L^{'}}) \leq [L^{'} : K^{'}] = p

as desired.

🔗

Exercises Exercises

1.

Complete the proof of Lemma 7.4.4 by showing that if

| p |_{v} = p^{- n}

for some positive integer

n,

then

[o_{K_{v}}^{*} : (o_{K_{v}}^{*})^{p}] = p^{n + 1} .

(Remember that

K

is a number field containing a primitive

p

-th root of unity.)

🔗

Hint.

Using the logarithm map, we obtain an isomorphism

o_{K_{v}}^{*} / μ_{K_{v}} ≅ Z_{p}^{n}

(even when

p = 2

). See [38], Proposition II.5.7 for details.

🔗

2.

Let

K

be a number field. Prove that for every positive integer

n,

C_{K}^{n}

is the intersection of the norm groups

{Norm}_{L / K} C_{L}

over all abelian extensions

L / K

of exponent

n .

🔗

Prev Top Next