Artin's primitive root conjecture for quadratic fields

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

H

ANS

R

OSKAM

Artin’s primitive root conjecture for quadratic fields

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 287-324

<http://www.numdam.org/item?id=JTNB_2002__14_1_287_0>

© Université Bordeaux 1, 2002, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

primitive

conjecture

for

_quadratic

fields

par HANS ROSKAM

RÉSUMÉ. Soit 03B1 fixé dans un _corps

quadratrique

K. On note S l’ensemble des nombres

_premiers

p pour

lesquels

03B1 admet un ordre

maximal modulo p. Sous

G.R.H.,

on montre que S a une densité.

Nous donnons aussi des conditions nécessaires et suffisantes pour que cette densité soit strictement

_positive.

ABSTRACT. Fix an element 03B1 in a

_quadratic

field K. Define S as

the set of rational

_primes

p, for which 03B1 has maximal order

mod-ulo _p. Under the

_assumption

of the

_generalized

Riemann

hypoth-esis,

we show that S has a

_density.

_Moreover,

we

_give

_necessary

and sufficient conditions for the

_density

of S to be

_positive.

1. introduction

In

_1927,

in a conversation with Helmut

_Hasse,

Emil Artin made the

following conjecture

[1,

preface;

2,

page

476]:

for any

integer

a, not

equal

to :f:1 or a _square, the natural

_density

of the set of

_primes

_p for which a is a

_primitive

root modulo _p inside the

set of all

_prime

numbers exists and is

_positive.

Artin based this

_conjecture

on the observation that a is a

_primitive

root modulo _p if and

_only

if for all

_primes

_l,

the

_prime

_p does not

_{split completely}

in

_{Here (I}

denotes a

_primitive

1-th root of

_unity.

_By

Chebotarev’s

_density

_theorem,

the

_density

of the set of

_primes

that do

_{split completely}

in

_{Q(~~, ~ a /(~}

is

equal

to

_{[Q ((I, ~qa :}

_(a~-1.

_Using

the

_principle

of inclusion and

_exclusion,

one

_expects

the limit

(1)

to be

_equal

to

It is not clear from this

_expression

whether the

_density

is

_positive.

_However,

one can show that the infinite sum is

equal

to a

positive

rational

multiple

A of Artin’s constant

In 1967

_Hooley

_[5]

_proved

that the

_density

₍₁₎

is indeed

_equal

to the sum

(2),

if the

_generalized

Riemann

_hypothesis

_(GRH)

holds true.

_Furthermore,

he

_explicitly

determined ca in terms of the

prime

factorization of a. Over

_arbitrary

number

fields,

there are _{two ways} in which Artin’s

con-jecture

can be

generalized.

We fix a number field K with

ring

of

integers

0 and a non-zero element a E C~ which is not a root of

_unity.

If we

replace

Q by K

in the above

_discussion,

we

_expect

the

following

generalization

to

hold: the set of

_{primes p of K for}

which a

_generates

(d/~pC~)*

has a

density

’

inside the set of all

_primes

of K. Note that is indeed a

_cyclic

group for all

primes

p.

Moreover,

the situation is

highly

similar to the rational _case, as the set of

_primes

_{p of K} for which

_(0/pO)*

is

_isomorphic

to

_(Z/pZ)*

has

_density

1. In 1975 Cooke and

_Weinberger

_[3,

see also

12]

proved

that if GRH

_holds,

this

_{generalization}

of Artin’s

_conjecture

is indeed

true and the

_density

is

_{given by}

₍₂₎

with

_Q

_{replaced by}

K. Lenstra

_[4]

_gave

a finite criterion to decide whether this

density

is zero. Note that we can

force the

_density

to vanish

_{by choosing}

an

_algebraic

_integer

_a, and

_defining

K to be

_equal

to for some

_prime

l.

The second

_{generalization}

of Artin’s

_conjecture

is in a ’rational’ direction:

the set of rational

_primes

_p for which the order of a in is

equal

to the

_exponent

of this _group, maximal for

_short,

has a

_density

inside

the set of all rational

_primes.

In this _paper we will _prove

that,

modulo

the

_generalized

Riemann

_hypothesis,

this

_conjecture

holds for

_quadratic

fields K. The reason for

_restricting

to

_quadratic

fields is twofold. As the

exponent

of

_depends

on the

_splitting

behaviour

_{of p}

in

K/Q,

a

proof

of the

_conjecture

needs to

_distinguish

between

_separate

cases, one for

each

_splitting

_type.

_Quadratic

fields admit

_only

two unramified

_splitting

types,

and

_already

in this case a fair amount of non-trivial Galois

theory

is needed to deal with the inert

primes.

More serious however is the fact that the

_analytic

_part

of our

proof

of the

conjecture

in the

quadratic

case

does not

_{readily generalize}

to

_higher

_degree

fields. Such a

_{generalization}

to fields of

_degree

k &#x3E; 3

_implies

that the set of

_primes

_dividing

a

’generic’

k-th order linear recurrent _sequence has a

_positive

lower

_density

[9].

Fix a

quadratic

field K with

ring

of

integers

0 and a E K*. The

_generic

primes

are

_by

definition those odd rational

_primes

_p that are unramified

in and for which there are no

primes

in d above _p in the fractional

ideal factorization of aO. As we are interested in

_density

_questions,

we can

For a

_generic

_prime

_p, the

_image

of a in is well defined.

If p

is a

_generic

_prime

that is inert in

_K/Q,

the _group is

_cyclic

of order

_{p2 -}

1. For these

_primes

we can ask whether a is a

_generator

of

( /pO)*.

We define the set S- as

S- = lp

generic prime : p

is inert in

_K/Q

and

_(a)

=

(OlpO)*I.

If _p is a

generic

prime

that

splits

in

K/Q,

the _group is no

_longer

cyclic,

but

_isomorphic

to

_F~

x

_Fp.

This _group has

_exponent

_{p -1,}

so we

define the set

S+

as

S+

_{= {p}

_generic

_{prime :}

p

splits

in

K/Q

and a has order

p-1

in

Before we state theorem 1 we make two more remarks.

1. The

_{generalization}

of Artin’s

_conjecture

that was

_{proved by}

Cooke and

Weinberger

does not

_imply

the existence of the

_density

of

_S+;

it is

_possible

that _p is in

S+

while a does not

_generate

_(0/pO)*

for either of the

_primes

p above p.

2. We _say that a and its

_conjugate

1ii are

multiplicatively independent

if

the

_subgroup

_{{a, 1ii)}

C K* is free of rank

_2,

or

_{equivalently,}

if the _map

Z2

--~ K*

_sending

(a, b)

to

aaab

is

_injective.

Theorem 1. Let K be a

quadratic field.

and

_fix

an elements a E K* .

De-fine

the sets

S+

and S- as above and _suppose the

generalized

Riemann

hypothesis

holds. Then

S+

and S- both have a natural

_density

inside the

set

_of

all rational

_{primes. Moreover,}

there exist rational numbers

_c+

and

c.,

depending

on _a, such that

We are not able to prove the existence of the

density

unconditionally.

Even in the case of Artin’s

original conjecture

it is not known

whether,

for a

given

non-square

integer

-l,

the set of

primes

p for which a is a

_primitive

root modulo _p is infinite.

To

_explain

the

_significance

of the rational numbers

_ct

and

_c~,

we first

sketch the

_proof

of the theorem. In section 2 we characterize the

_primes

in

S+

and S- in terms of

_splitting

conditions in the fields

where I _ranges over the

_prime

numbers. The fields

Ll

are the

analogues

of

To _express the densities as infinite sums similar to

_(2),

we

_adapt

_Hooley’s

arguments

to our situation. This is

straightforward

for the set

_S+;

for the inert

_primes

additional

_arguments

are needed. In order to _prove that these

infinite sums are

_equal

to the Euler

_products

in theorem

_1,

we need to know the obstruction for the fields with 1

_ranging

over the

_primes,

to be

linearly disjoint

over K. Here and in the rest of this _paper, we _say that a collection of subfields

fLiliEI

of some field

L

is

_linearly

_disjoint

over a subfield K~ of

_L,

if the

_following

holds: for

_{all j}

E

_I,

the intersection of

_Lj

and the

_compositum

of the fields is

_equal

to K. In

_addition,

we

have to

_compute

the

_degrees

of the fields

_Li.

In

_propositions

8 and 10 we

determine a finite set of

_primes

I such that the fields are of

_{‘generic’}

degree

and

_{linearly disjoint}

over K. The rational numbers

c+

and

_c~

take care of the

remaining primes,

and can be seen as ’correction factors’.

In Artin’s

_original

_conjecture,

the

_density

of the set of

_primes

p for which

a is a

primitive

root vanishes if and

_only

if a is -1 or a _square, the

if-part

being

obvious. In our situation there are also some more or less obvious

situations of zero

density.

Proposition

2. Let

_K,

_a,

S+

and S- be as in theorem 1.

a. In each

_of

the

_following

_cases, the set

S+

is

_finite:

_(i)

a is a root

_of

unity,;

(ii)

a is a _{square in}

_K*;

(iii)

K =

Q((3)

and _a is _a cube in K*.

b. In each

_of

the

_following

_cases, the set S- is

finite:

(i)

a and a are

multiplicatavely dependent;

(ii)

a is a cabe in

K*;

(iii)

a

square in

Q*;

(iv)

K =

Q((3)

and 2 is

a cube in K*.

Proo f .

a. If a is a root of

_unity,

the set

S+

is

_clearly

finite. Now assume

that a is a _square in K* and that _p is a

_{generic prime}

that

_splits

in

K/Q.

As p is

odd,

the

exponent

of

_(O/pO)*

is even, so the order of a in this

group divides

(p -1)/2.

We find

p g

S+ and

the set

S+

is

empty.

If K is

_equal

to

_Q((3)

and _p

_splits

in

_K/Q,

the

_exponent

_{p -1}

_{of the group} is divisible

_by

3. In case a is a cube in

K*,

its order modulo

pO

is at most

_{(p -1 ) / 3}

and hence

S+

is

_empty.

b. Let _p be a

generic prime,

inert in

K/Q.

Assume that a and a are

multiplicatively dependent,

say

aaj,6

= 1 for

integers

a and b with a non-zero.

_Taking

the norm

to Q yields

= 1. In _case that a + b =

0,

we find aa to be

_rational,

so the order of a in

(C?/pn)*

divides

a(p 2013 1).

Otherwise,

we have

_and,

as

_{NK/ (a)}

is

_congruent

to

aP+l

modulo

_pO,

the order of a in is at most

_2(p

+

_1).

The set S- is

clearly

finite in both cases.

For a

_generic

_prime

3 that is inert in the order

_{p2 -1}

of the

cyclic

group is divisible

by

3. If a is a cube in

_K*,

its order in

If a has order

p2 -1

in

(OlpO) *,

we find that =

aP+1

mod

_pO

has order

_{p -1}

in

_Fp.

As _p is

_odd,

this

_implies

that

_NK/Q(a)

is not a _square

in

_Q*.

In other

_words,

if

_NK/Q(a)

is a _square in

_Q*

then ,S- is

_empty.

Finally

assume .~ is

_equal

to

_(~(~3)

and _p is a

_{generic prime}

that is

inert in so _p = 2 mod 3. Write

a/a

= ,Q3

for some

{3

E K*. As

{3

has norm

_1,

its order modulo

pO

divides _{p + 1} and hence the order of

a/a

mod

_pO

divides

_{(p + 1)/3.}

_Using

_{the congruence}

_{a/a -}

mod

_pO

we find that the order of a

mod pd

is at most

(P2 - 1) /3,

hence S- is

empty.

D

Apart

from the cases listed in

_proposition

_2,

there are other situations

in which one of the densities vanishes. To state

_them,

we let D be the

discriminant of K and we assume that a is not a cube in K*. This last

assumption implies

that at least one of the elements or

a/a

is

not a cube in

_K*,

and the field

_k3

below is well defined. For each

_{prime 1}

define the

_{field kl}

as follows:

For

_{K 0}

_Q((3)

these are

_quadratic

fields. In section 6 we _prove the

follow-ing

theorem.

Theorem 3. Let

_K,

a,

S+

and S- be as in theorem 1.

a. The set

S+

has

_density

_0,

and is

_{actually finite, if}

one

of

the

_following

statements holds:

i. a

_satisfies

one

_of

the conditions

of proposition

_2a;

ii. K =

Q(J5),

_a is _a 15-th

power in K* and is

equal

to the

maximal real

_{subfield of}

b.

_Define

the

_{fields k2,}

_k3

and

_k5

as above. The set S- has

density

0,

and

is

_{actually finite, if}

one

of

the

following

statements holds: i. a

_satisfies

one

_of

the conditions

_{of proposition 2b;}

ii.

_Among

the

_{fields K, k2,}

_k3

and

_k5,

there exist three

_{different fields}

whose

_compositum

has

_degree

4 over

Q.

If

the

_generalized

Riemann

_{hypothesis holds,}

these are the

only

cases in which the

_{density of}

S+

or S- is zero.

We have chosen not to

_give

an

_explicit

formula for

ct

and

_{c. ,}

as such a

formula will be

_complicated

and not very

_{enlightening.}

To

get

a

feeling

explicit examples

in section 7. We _gave

_precise

formulas for the

_density

of related sets in

_[8].

There is an infinite set of a’s for which the constants

_ct

and

_ca

are

easily

computed. Hooley proved

that in the

_original

_conjecture

the

_equality

_ca =1

holds if and

_only

if a is not a _power in

Q*

and the

prime

2 is ramified in

We have the

_following

similar result.

Theorem 4. Let K be a

quadratic

field

with

_composite

discriminant and

let _p’K be the torsion

_subgroup

_of

K* . We assume that a and ä are

mul-tiplicatively independent,

that _a,

_1ii)

is torsion

_free

and that the

prime

2 is

_totally

_ramified

in

_L2

=

If

GRH

holds,

then the

densities

_of

S+

and S- are

_{given by}

the

_following

formulas:

Independently,

Yen-Mei J. Chen and

_Jing

Yu have

_{recently proved}

_that,

modulo

_GRH,

the set S- has a

density

for a restrictive set of

imaginary

quadratic integers

a

(private

_{communication,}

unpublished).

For some _very

specific

a’s,

they

showed that this

_{density equals}

a

positive

rational

multiple

of an Euler

product

as in theorem 1 and 4.

The _paper is

_organized

as follows. In section

_2,

we characterize the

_primes

in

S’+

and S- in terms of their

_splitting

behaviour in the extensions

_LI/Q,

where I _ranges over all

primes.

These fields

ILI JI

tend to be

_{linearly disjoint}

over K. To make this

_precise,

we

_study

the

_{composita L~}

of the fields

in section 3. Here and in the rest of the _paper, the index variable 1 will

_always

_range over

_prime

numbers . Section 4 contains the

_analytic

part

of the

_proof

of theorems 1 and 4: if GRH holds then both

S+

and S-have a

density

which is

given

by

an infinite sum similar to

_(2).

The

_proof

of the theorems 1 and 4 is

_completed

in section 5 and theorem 3 is

_proved

in section 6.

_Finally,

in section 7 we

_compute

for some a’s the rational

constants

_c;

and

_{ca .}

2.

_Splitting

conditions

In this section we characterize the

primes

in

S+

and S- in terms of their

splitting

behavior in certain number fields. The order of a modulo a

_generic

prime

p is non-maximal if and

only

if the there exists a

_prime

1 such that

(3)

1 divides and a is an l-th _power in

Namely,

if I satisfies

₍₃₎

then I divides the

_exponent

of

_(0/pO)*

and a has

that the order of a modulo

_pO

is non-maximal.

If p

is inert in

K/Q,

the

group is

cyclic

and any

prime

divisor I

I

(a)]

satisfies

(3).

If p splits

in

_K/Q,

the _group is of

_{exponent p 2013}

1 as it is

isomorphic

to the

_product

of two

_cyclic

_groups, both of order _{p - 1.} As a

is non-maximal modulo _p, there exists a

prime I

_{I p - 1}

such that the order

of a in divides

(p -

1)/l.

Therefore,

a is an 1-th power modulo

both

_primes

_pip

in K and 1 satisfies

_(3).

If we define the sets

{p

generic

prime,

inert in

_K/Q

and

_{either 1}

_{# (O /pO)*}

or

(OlpO)*ll

and

{p

generic

prime,

splits

in and

_{either I ( #(O/pO)*}

or

where denotes the

_subgroup

of 1-th _powers in we find

the

_{following equalities:}

-

r---We can reformulate statement

₍₃₎

as the

complete splitting

of

Xi -

a

modulo the

_prime(s)

in d above _p. The set of these

_prime(s)

is stable under the non-trivial

_automorphism

of K.

_Therefore,

if a

splits completely

modulo these

_primes,

so does

Xi -

a. Here a denotes the

_conjugate

of a over

Q.

This

_gives

the

following

characterization of the

_primes

which are

not in

_Sa

or

st.

Proposition

5. Let 1 be a

_prime

and

de, fine

the

field

_Li

as

Lt = K(~l ~ ‘ a~ ‘ a) ~

a. For a

generic prime

_p that

splits

in

K/Q

the

following

equivalences

holds:

p 0

Si

4====~ _p

_{splits completely}

in

b. For a

generic prime

_p that is inert in

K/Q

the

following

equivalence

holds:

p 0

Si

~

_{pd splits completely}

in

_LIIK.

By

Chebotarev’s

_density

theorem and this

_proposition,

we conclude that the

sets

_Si

and

_Sl

have a

_density

in the set of all

_primes.

The same conclusion

holds for the finite intersections

_Sn

=

n E as the

_primes

in these sets are characterized in terms of their

splitting

behaviour in the number field

_Ln,

the

_compositum

of the fields Because of the

equalities

(4)

and

_(5),

the

_primes

in

S+

and S-are characterized

by

splitting

conditions in the infinite extension

Loo?

the

compositum

of the fields

Li

where 1 ranges over all

_prime

numbers. One

would like to prove the existence of the densities of

S+

and S-

by

applying

a theorem

analogous

to Chebotarev’s

density

theorem to the field

Loo.

However,

even the formulation of such a theorem is

non-obvious;

all

primes

of

_Loo

which do not lie above 2 are ramified and therefore do not have a

well-defined Frobenius element.

Our

_proof

of the existence of the

density

of

S+

and S- is

_analogous

to

Hooley’s

proof

of Artin’s

_original

_conjecture.

The

_key

observation is the

following.

As

S+

can be seen as a ’limit’ of the sets

_{S~ ,}

one

_expects

the

density

of

S+

to be

_equal

to the limit of the densities of the sets

_S¡.

A similar observation holds for the set S- . In section 4 we will

_{adapt Hooley’s}

arguments

to our

situation,

and _prove that this limit

_argument

is indeed

valid if we assume GRH. This is

_{straightforward}

for the set

_S+,

but causes some difficulties for the set S-.

Proposition

5 characterizes the

_primes

_p in

Sl

by

the

_property

that the Frobenius class of the

_prime

of K in

_{Li /K}

is non-trivial. In order to

_{adapt Hooley’s analytic}

_arguments,

we have to

characterize the

_primes

in

_S~

as those

primes

which have a non-trivial

Frobenius in extensions of

_Q.

Proposition

6.

_Suppose

_p is a

generic

prime,

inert in

K/Q.

Define for

each

_{prime 1}

the

a. Let 1 be an odd

_prime

and assume that K is not the

quadratic subfield

of

Q((,).

Let

_N2

be the

_unique

extension

_of

_{Q ( f t}

+

~j 1 )

that contains neither K

_{nor (i}

and such that

_KN2

=

K ( ("

The

fields

N1

The

_{following equivalence}

holds:

(6)

Si

4=* p

splits completely

in either

_Nl/Q

or

Ni

/Q.

b. Assume

_NK/Q(a)

is not a _{square in}

_Q*.

_IfNK/Q(a)

is a _{square in}

K*,

then _p is in

_{S2 .}

_If

_NK~Q(a)

is not a _{square in}

K*,

then the

following

equivalence

holds:

p rt

S2

~? _p

_{splits completely}

in

_N2’/Q.

Proof.

We first

_give

a different formulation of

_proposition

5b in case

p :A

I.

Fix a

_{prime I,}

and let I be a

generic

prime

that is inert in

K/Q.

All

primes p

above _p in

_Li

are

unramified,

so their Frobenius

_{automorphisms}

(p,

Ll/Q)

are well-defined. The order of

(p,

Li/Q)

is

equal

to the residue class

_{degree of p}

in

_LI/Q

and

_independent

of the choice of _p above _p. We reformulate

_proposition

5b for an inert

prime p =A

I as follows:

a. Let 1 be an odd

_prime.

Because of the

_{equivalence above,}

we first

characterize the elements in of order 2 that are non-trivial on

K. Assume _p is such an element. The restriction also has order 2

and therefore lies in the maximal

_exponent

2

_subgroup

of

_{Gal(K((,)/Q).}

Here

_(t

denotes a

_generator

of the maximal real subfield

of

_Q(~~).

Let T and 0’ denote the

_generators

of and

respectively.

The

_{assumption K 0}

_Q((,)

_implies

that

Gal(K(~~)/fa(~‘ )) =

(Q)

x

_~T)

is

_isomorphic

to Klein’s four _group. As _p is non-trivial on

_K,

we find that

_pIK((,)

_equals

or aT. Fix an element

p

E

_(a,

and define

_Fp

as the fixed field of

_p.

Consider the

following

exact sequence:

The _group

_{Gal(Ll/ K((l»}

has odd

_exponent,

hence _any element of order 2 in

_Gal(LI/Fp)

restricts to

_p.

This proves that the sequence

splits

and that

there indeed exist elements _p E

_Gal(LI/Q)

of order 2 that are non-trivial on K. We fix an

isomorphism

The abelian _group G =

Gal(LlI K((l»

is _a module _over the

group

ring

F~~(p)~.

Because I is

_odd,

there is a

decomposition

G =

Hp

x

_{HP ,}

with The element

_p

acts

_trivially

on

Hp

and

_by

inversion on

_Hi.

We conclude

By

definition of

_Hi,

the _group

_(p))

is the maximal

_quotient

of

G that is abelian of

_exponent

l. In

_particular,

_{Hi )4}

_(p)

is a characteristic

subgroup

of G and therefore normal in

Now we are able to _prove the

_equivalence

_(6).

Define the normal fields

Nl

and

_N2

as the fixed fields of

H;

)4

(0’)

and

_H~T

)4

_{respectively.}

Let

_{p # I}

be a

_generic

_prime

that is inert in and choose a

_prime

_p

above _p in

_Li.

_By

the

_equivalence

₍₇₎

and the characterization

_(8),

we find

that

_{p ~}

_ST

if and

_only

if the Frobenius

_{automorphism of p}

in lies either in

_H;

»

_(a)

or in )4

_{(aT) .}

In other

words,

the

_prime

_p is not in

Si

if and

only

if p

splits

completely

in either or

generic

and inert in

_K/Q,

then 1 does not divide

_~(d/dC~)*

and is therefore contained in

_ST.

On the other

_hand,

the

_{prime I}

ramifies in both and

_N,2/Q,

hence does not

_{split completely}

in either of these extensions.

To determine the fields

_Nl

and

_N,2,

we have to understand the

conjuga-tion acconjuga-tion of a and aT on

Gal(L,/K((,».

This can be done

by considering

the Galois

_equivariant,

bilinear and

_{non-degenerate}

Kummer

_pairing:

Using

the fact that a acts

_{trivially on (I}

and

_interchanges

a and

_a-,

we

find for h E the

_following

_{equivalences:}

Therefore,

the group

H§

corresponds

to the field

_{K(~l, ~ NK~Q(a)).}

_Taking

the invariants under a

yields

Nl

=

Q((1 ,

The fixed field

_{K(~c, ~ a)}

of

_H~.

is a

_quadratic

extension of

N2.

As err is

non-trivial on both K and

(1,

the field

N,2

does contain neither of them and

KN2

=

K(~~, ~ a).

On the other

hand,

_any extension N of

Q((#)

that

contains neither K

_{nor (I}

and for which

_{~K(~~, ’ a) :}

_N~

= 2 is an

_exponent

l extension of

_FaT

and hence

_equal

to

_NZ.

This concludes the

_proof

of 6a. b. Assume p is a

generic prime,

inert in The

degree

of

L2

=

K (Va,

divides 8. If

_{NKIIQ (a)}

is a _square in K* but not a _square in

Q*,

then

_L2/Q

is

_cyclic

of

_degree

4. The Frobenius of

_{all pip}

in have order 4 and the

_equivalence

₍₇₎

_implies

that _p is an element of

_{S2 .}

If

_N(a)

=

NK~Q(a)

is not a _{square in}

K*,

then

L2/Q

is dihedral of

degree

8,

and the Galois _group of both

_L2/K

and

_{L2/Q( N(a))}

is

isomor-phic

to Klein’s _{four group.}

_{If p splits}

in

_{all pip}

in

_L2

have a

Frobenius of order 2 and p is not an element of

S2.

On the other

hand, if p

is inert in

_K/Q

and in

_{Q( N(a))/Q,}

it

_splits

in the third

_quadratic

field

Q(ýD. N(a»

C

K( N(a)),

where D denotes the discriminant of K. As

L2 I Q ( Ý D . N ( a»

is

_cyclic

of order 4 and the

_primes

above _p are inert in

N(a)),

the Frobenius of

_{all p I p}

in

_{L2 /Q}

have order 4

and p is in 82 .

0

The

_proof

of

_proposition

6

_yields

the

_following

_corollary,

which we need in

the

_proof

of

_proposition

19 in section 6.

Corollary

7. Let 1 be an odd

_prime

and let _p E

_Gal(Li/(a)

be non-trivial on K such that

_pIK«I)

has order 2. Furthermore

_define

the set

If

K is not contained in

_Q(Cl)

then the

_{following equivalence}

holds:

where

_Ni,p

is the

_unique

_field

in

_{N~ }}

that contains the

_{fixed field of}

PIK((,).

The set

_Cl

has

_cardinality

Proof.

We use the notation from the

proof

of

proposition

6. Let p E

be non-trivial on

_K,

and of order 2 when restricted to

_K((I).

By

construction

_exactly

one of the fields

Nl

and

Nl

contains the fixed field

of so is well defined. In the

proof

of

proposition

6 we saw that

p has order 2 if and

only

if p is trivial on

Nl

or As K is contained in the

_compositum

of these

_fields,

the element _p can not be trivial on

both,

and the

_equivalence

is

_proved.

To

_compute

_c(l),

we use the characterization

(8)

of elements in

Cl

and

find

_{c(l )}

=

degrees

of

_L,

over

KNI’

and

KN,2

_{respectively. Using}

the definitions of

N1

and

_N2

_yields

the desired result. 0

By

Chebotaxev’s

_density

theorem and

_proposition

5 we find for each

_prime

I the

_following

formulas:

Here

_c(l)

denotes the

_cardinality

of the set

_Ci,

which was defined in

corollary

7. In the

_proof

of theorem

_1,

we need to know the values of these densities. It turns out that for all but

_finitely

_many

_{primes l,}

these densities have a

_{generic description}

in terms of l.

Proposition

8. Let r be the

free

rank

of

(a, ix)

and let t be the order

_of

the torsion

_{subgroup of}

_K*/(a,

_d)

-a. In each

_of

the

_following

cases

I(S/)

is

_positive:

(i)

1 &#x3E; 5

_prime;

_(ii)

1 =

3

_{and K 0}

_{Q(~3); (iii) 1}

= 2 and

K*2.

For all

_{primes 1 f}

2t such

_{that K 0}

_Q(,)

we have

[Li : K] = (l

-

1)l’’.

b. In each

_of

the

_following

cases is

_positive:

(i)

1 &#x3E; 5

_prime;

_(ii)

1 =

3,

K ~

Q ((3)

K*3;

(iii) 1

= 2 and

Q*2.

If r equals 2

2t is a

_prime

such that

K 0

Q(~1),

then zue have

and both

_Nll

and

_N2

are

of

degree

l(l -

1)

over

Q.

Proof.

a. As is contained in

Li ,

we have

[Li :

K~ &#x3E;

[7~(~) :

K~ &#x3E;

2

for all

_{primes 1}

&#x3E; 5. The same

_inequality

holds for 1 =

3,

provided

K is

not

_equal

to

_Q((3).

If a is not a _square in

K*,

the extension

L2/K

is also

of

_degree

at least 2. In all these _cases, the set

_st

has

_positive

_{density by}

formula

_(9).

The fact the t is finite is well known. Let 1

_~’

2t be a

prime

such that

K is not contained in

_Q(Ci).

The last condition on 1

_implies

_{[Li : K] =}

[Li :

K] =

(l

-

K((,)].

By

Kummer

_theory,

the

degree

[Li : K(Cl)]

equals

the

_cardinality

of the

_image

of W =

(a, a)

in

First we show that the _map

is

_injective.

_Namely,

if x mod

K*l

is in the

_kernel,

we can write x

= y~

for _{some y} E

_K((,)*.

_Applying

the norm _map from to K

yields

the

equality

= We find that x is trivial in

K*IK*I,

as

its order divides both 1 and

_{~K(~~) : K~.}

If we can write w =

y’

for some w E W

and y

E

K*,

the

_image

of _y in is an 1-torsion element. As we assumed that I does not divide the

order of the torsion

_subgroup

of

_{K*/W ,}

we find that _y is in W and the

map

is

_injective.

With these two

_{observations,}

we find that the

_degree

[L3 : K((l)]

is

equal

to the index

_{[W :}

_Wl].

As K is

_{by assumption}

not contained in

and 1 is

_odd,

the _group W has no non-trivial 1-torsion. Therefore we have

[W :

= F and _we conclude that

_Li

is of

_degree

(l - 1)1’

_over K.

b. Let I be an odd

_prime

and assume that K is not contained in

_{Q ((I) .}

By

corollary

7 we have

This

_implies

the

_inequalities

e 1 2

for 1 &#x3E; 5. In case a is not a cube

in

_K*,

at least one of the elements

_{or g}

is not a cube in K* and we find the _upper bound

c(3)/[L3 : K~ 2/3.

Using

formula

(10),

we see

that in these cases

_Sl

has

_{positive density.}

The result for

b(,S2 )

follows

immediately

from

_proposition

6b.

If a and 1ii are

_{multiplicatively independent}

and the

_{prime I}

does not

divide

_2t,

forms an

Fi-basis

for

by

the

proof

of

_proposition

8a. In

_particular,

neither alii

_{nor g}

is an l-th _{power in}

K ((I ) *

and

_by

Kummer

_theory,

K«(,,1NK/Q(a»

_{and KN2 =}

K(~l, ~ a )

are of

degree lover

K ((i ) .

_Substituting

this in the above

_equality

yields

the result for

_K].

The field K is

_{linearly disjoint}

from both

Nl

sowefind

_{[Nl : :(~~=[KN~}

_{:K~=l~~K(~~):K~=Z(l-1).}

The same

equality

holds if we

_replace

N~

by

N2.

To conclude the

_proof

of

_{proposition 8b,}

we have to _prove &#x3E; 0 in case K is contained in

Q((/)

for some

_{prime ~5.}

This is left to the

3. The Galois structure of

_L~/Q

As we

_already

noted in the last

_section,

the finite intersections

Sn

=

nii.

Si

have a

_density.

These can can be

computed by

Chebotarev’s

_{density theorem,}

in the same _way as we

_computed

and

6(Si )

in

₍₉₎

and

_(10).

_Using

the

_principle

of inclusion and

_exclusion,

we

find the

_following

formulas:

with

_Ld

the

_compositum

of the fields and

_L1

=

K,

and

with

_{c(d) _}

E

_{Gal(Ld/Q) :}

id

and Q2

= In the _next

section,

we will _prove that if GRH is

_true,

these formulas also hold if we

take the limit n = I _-

oo,

thereby proving

the existence of

6(S+)

and

_6(S-).

It is not clear from

₍₁₁₎

and

_(12),

whether these densities are

positive.

If the fields are

_{linearly disjoint}

over

_K,

the summands

of

₍₁₁₎

and

₍₁₂₎

are

multiplicative

in d and we obtain

product expressions

for the densities of

_Sn

and In this case it is _easy to conclude whether these densities vanish.

_However,

the fields are not in

_general

lin-early disjoint

over K. For

_example,

if K =

_{Q( -15)}

the field

Q( v’5,

3)

is contained in both

_L3

and

_L5.

Proposition

10 below shows that the fields

_ILIJI

are not too far from

being linearly disjoint

over K. We will need the

_following

lemma.

Lemma 9. Let

_F/Q

be a

finite

abelian extension

_of

_exponent

e.

_{If n}

and m are

_relatively

_{prime integers}

then also

_of

_exponent

e over

_Q.

Proof.

As F is contained in the

_{intersection,}

it is sufficient to _prove that the

_exponent

of n divides e. Recall that the character _group

X (L)

of an abelian number field L is defined as the _group of

homomor-phisms

Gal(L/Q) -

C*,

and that

_X(L)

is

_{(non-canonically)}

_isomorphic

to

Gal(L/Q).

It is therefore sufficient to _prove that the order

_{of 1/J}

divides e

for

_{all 1/1}

E

_X(F((n)

_{As 1/1}

is both in

_X(F(Cn))

and in

_X(F(m»

we can

_{write 1/1}

=

Xlpl

and 1/J

= _X2P2 with

Xl E X2 E

X(Q(m»,

and _{pl, p2 E}

_X(F).

The element

_{pi lp2}

=

XlX21

is in

X(F),

so its order

divides e. As and

Q((m)

are

linearly disjoint

over

Q,

the _group

is

_isomorphic

to

_X(Q((n»

x

X(Q((m»

and the order of _Xl di-vides that of

_{Consequently,}

the order of

_0,

which divides the least

For _any number field

_F,

we denote

_by

Fab

the maximal subfield of F that

is abelian over

Q.

For

integers

k and _n, we denote their

_greatest

common

divisor

_by

_{(k, ~t).}

Proposition

10. For all

_{positive integers}

_n, let

_Ln

be the

_compositum

_of

the

_fields

_{De fine}

the

_integer

_f

as

follows:

For all

_positive

_squarefree

_integers

n and m that are

_relatively

_prime,

the

following

statements hold:

d.

_Ln

_{n Lm}

is abelian over

_{Q of exponent dividing}

_e, with

e.

_L~

_{fl L2 f,}

_{if nrrL}

and

_{2 f}

are

_{relatively prime.}

f.

_{Ln fl Lm}

_{= K, if}

the

_prime

2

_{ramifiées completely}

in

_Lab/Q.

Proof.

For a

prime 1,

the field

_L,

is defined

by

and 10a

follows

_immediately.

The

_degree

_L~)]

divides both

_{[Ln :}

and

_{[Lm :}

L~].

As

_Lab

contains

_K(~k)

for each

_{squarefree integer k,}

we find that

~Ln :

L’b]

and

_{[Lm :}

divide

n2

and

_m~,

_{respectively. Using}

the

_assumption

that n and m are

relatively

prime,

this _proves lOb.

In order to _prove

_10c,

we first show that for an odd

_{prime 1}

the field

Lib

coincides with unless both K is

_equal

to

_(~(~3)

and 1 is

_equal

to 3. The extension

_LlIK((l)

is a Kummer extension of

degree

_dividing

12.

The

intermediate fields of this extension are of the form

K((l,

~),

where W is a

subgroup

of

(a,

a),

the

multiplicative subgroup

of K*

generated by

a and

a. If

_Lib

is

_strictly

_larger

than

_K(~l},

there exists a

_{3

E

_{(a, a),}

not an 1-th

power in

K(~~}*,

such that

~"(~,~9)

is abelian over

Q.

As a _consequence,

the 1-extension is

_normal,

which forces a

primitive

1-th root of

unity

to lie in

_{K (~) .}

Because 1 is

_{relatively prime}

to the

_degree

over

_,K,

the 1-th roots of

_unity

are contained in If K is not

_equal

to

Q (~3 )

or 1 is not

_3,

this is a contradiction and we find

Lab

=

K((l).

For

K =

Q(~3}

_{we use} the Kummer

_pairing

_{as in the}

_proof

_of

proposition

6 and find that

_L3b

is

_equal

to

_Q(~3,

with W the

_{largest subgroup}

of

(a, 1ii)

on which

Gal(Q((3)/Q)

acts

by

inversion. This

yields

the

equality

L3b

=

Q((3,

~~),

a

cyclic

extension of

Q

of

_exponent

dividing

6.

For the

_general

_case, we abbreviate the

compositum

of the fields

by Mn,

which is

_clearly

contained in

_{Lg .}

_Restricting

_{automorphisms gives}

the

_following

_injective

map:

Injectivity

follows from the fact that the fields

_{generate L,~.}

As the intersection

_Labn

_Li

_equals

_Lab

for t

₁

_n, the

_image

of the

_subgroup

C

_Gal(Ln/Mri)

maps

surjectively

to each

component.

For

different

_primes

the _groups have

_relatively

_prime

_orders,

as

is an 1-extension for all L

Consequently,

the _group

maps

surjectively

to the

_product,

the

_{map 0}

is an

_isomorphism

and

Lab

equals

Mn.

This _proves 10c.

The number e in lOd is well defined.

Namely,

if K is

equal

to

_Q((3)

then

_Lab/Q

is of

_{exponent 2;}

if

_L2b

would be

_cyclic

of

_degree

4 over

Q,

its

unique quadratic

subfield K would be real. To _prove 10d we assume that

K is not

_equal

to

_{Q ((3),}

that m is odd and n _even; the other cases are

similar and left to the reader.

_Using

lOb and lOc we see that

Ln

fl

_L~

is contained in fl

_K((m)

C

Lab (cn) n

Statement 10d follows

by

applying

lemma 9 with F =

L2b,

_a field of

_exponent

2 or 4 over

_Q

As K is contained in

_L2b,

the conductor D of K divides

_{f .}

Note that if K is

_equal

to

_(~(~3),

then

_f

is a

multiple

of 3. Assume nm and

2 f

are

relatively

prime,

hence K is not contained in and the fields

_K((n)

and

_K(~m)

are

linearly disjoint

over K.

_Using

lOb and 10c we find the

equality

Ln fl Lm

= K. To _prove the second

equality

in

10e,

we first note

that the definition of

_{L2 f}

_{only depends}

on the

squarefree

_part

of

_{2f -}

With

this in

_mind,

we

_again

use 10b and 10c and find:

As n and

_f

are

relatively prime,

the field K is contained in

Q(Cf)

but not

in

_{Q (Cn)}

and an _easy

computation

shows that the

degree

of

K (Cn)

n

is 2. As K is contained in this

_{intersection,}

we find

_Ln

n

_{L2 f}

= K.

To _prove

_10f,

it is sufficient to prove the

equality

Ln fl L2m

= K,

for

_odd,

squarefree

and

_{relatively prime integers n}

and m. Assume that the

_prime

2 ramifies

_completely

in hence it ramifies in all subfields of

_Lab

Therefore,

the fields

_L2b

and

_Q((nm)

are

_{linearly disjoint}

over

_Q

_and,

as

K C

L ab 2

so are K and

Furthermore,

as 2 is unramified in

(~ (~’3 ),

equalities:

As K is a subfield of

Ln

fl

_L2m,

this proves the last statement of the

propo-sition. 0

Corollary

11. Let r be the

free

rank

of

~a, a~ .

There exists a

positive

constant

_depending

on _a, such that

for

all

_positive

square f ree

_integers

n the

_{following inequalities}

hold:

Assume that a and it are

_{multiplicatively independent}

and

define for

each

positive

sq2carefree

integer n

the number

_{c(n) =}

E

_{Gal(Ln/Q) :}

id, q2

=

id}.

For all - _&#x3E; 0 there exists _a constant _r-2,

depend-ing

on - and _a, such that

_for

all

_positive

_{squarefree integers}

n the

_following

inequality

holds:

I .

Proof.

The _upper bound for

_{[Ln : K~}

follows

_easily

from

_proposition

10a.

To obtain the lower

_bound,

let no be the maximal divisor of n

prime

to

2t f .

Here

_f

is defined as in

proposition

10 and t is the order of the torsion

subgroup

of

_{K* / (a, a) .}

Note

_that,

as n is

_squarefree,

the

_quotient

n/no

is

bounded

_by

_{2t f . By}

_{proposition 10e,}

the fields

_ILIIllno

are

linearly disjoint

over I~.

_By

Galois

theory,

the _group is canonical

_isomorphic

to the fibred

_product

of the _groups over

Gal(K/Q).

As a _consequence we find the

equalities

c(no) =

c(l)

and

[Lno : K]

=

K~.

Applying

proposition

8a

_yields

with xi =

independent

of _n.

Using proposition

10e and 8b we find that for each - &#x3E; 0 there exist

constants ~2 such that the

following

holds:

Using

proposition 10,

we are now able to describe

_Gal(Ln/Q)

in terms of the _groups

_{IGal(LI/Q)Ill..}

This will be used in the

_proof

of theorem 3.

Proposition

12. Let n be a

_{positive squarefree}

integer

and

define

6

e = 4

if

2 1 n

and

L2b/Q

is

of

_{exponent 4;}

2 otherwise.

2 otherwise.

Let M be the

_{compositum of}

the

_fields

where

_Ml

is the maximal

subfield of

Li

that is abelian over

Q

and

of

_exponent

dividing

e. Then the

fields

are

_linearly

_disjoint

over M.

Consequently,

Gal(Ln/Q)

is

the

_{fibred product of}

the _groups over

Gal(M/Q):

Proof.

Let l

_I

n be a

prime.

We have to prove the

equality

[MLI : At] -

M] =

[Ln : M].

By

Galois

theory,

the extensions in

the

_diagram

that are indicated

_by

the same

symbol,

have the same

_{degree. Therefore,}

it is

sufficient to prove that M is the

disjoint

com-positum

of Ll n M

_{and MnLn/z}

over

_LznLn/z.

The

_compositum

and

_MnLn/z

contains the fields

_Mq

for all

_{primes q I n}

and is therefore

_equal

to M.

_{By proposition 10d,}

we know that

_{Lz n Ln/z}

is abelian over

Q

of

exponent

dividing

e. As

Mi

C

Li

is the maxi-mal abelian subfield of

_exponent

e over

_Q,

this

intersection

_equals

_M,

n

_Ln/l

= M _n

Li

_n

-1 1

In other

_words,

the field M is the

_{disjoint compositum}

of

_L,

fl M and

M n over

L,

n

_Ln/l.

0

We conclude this section with a well known result that will be used in the

proof

of theorem 14.

Proposition

13. There exists a constant _x3,

depending

on _a, such that

for

all n E

_Z&#x3E;l

and all

_subfields

F C

Ln

the

_following

_inequality

holds:

where

_{dF is}

the absolute value

_of

the discriminant

_of

F.

Proof.

It is sufficient to _prove the formula for F =

Ln,

_as the

root-discriminant is maximal for this choice of F. Let R be the set

of rational

_primes

that

_ramify

in

_Ln/Q.

We write R =

Ri

U

R2,

with

Ri

the set of

_primes

in R that do not divide

_{[Ln : Q]}

and

_R2

=

RBR1.

If

K in the factorization of the fractional ideal

_(a).

The

_primes

in

_Ri

are

therefore bounded

_{independently}

of n. Let

be the ideal factorization of the different of

_L~/Q.

The

_integers

_ap are

independent

of the

_{primes p}

above _p, as

_Ln

is normal over

Q,

and

satisfy

the

_following

bounds

_[11,

_page

_58]:

ap =

ep -1

if _p is

_{tamely ramified;}

ap

+

_{epvp (ep)}

if p

is

_wildly

ramified.

Here _vp denotes the

_p-adic

_{valuation and ep is the ramification index of} p

in

_Ln/Q.

If _p is

_wildly

ramified in the Galois extension then _p divides

_{[Ln :}

_Q]

so that _p E

R2.

By taking

the norm of to

_Q,

we

find the

_following:

..

As the

_primes

in

Ri

are bounded

independently

of n and the

degree

[Ln : Q]

is at most

_by

_corollary

_11,

this _proves the

_proposition.

D

4. The

_analytic

_part

of the

_proof

of theorems 1 and 4

The

_proof

of theorems 1 and 4 is in two

_steps.

Theorem 14 below is the

_analytic

heart of the

_proof:

both

S+

and S- have a

density

if we assume GRH. As we mentioned in the

_{introduction,}

the

_proof

is

along

the

lines of

_{Hooley’s proof}

of Artin’s

_original

_conjecture

_[5,

see also

7].

In the

next section we finish the

_proof

of both theorems: the formulas for the

densities in theorem 14 below are

equal

to Euler

_products.

This

_part

is

more

_algebraic

and does not

_require

GRH.

Theorem 14. Assume the

_generalized

Riemann

_hypothesis

holds.

_If

a is

not a root

_of

_unity,

then the

_{density of}

S+

exists and

_equals

If

a and its

_conjugate

d are

multiplicatively independent,

then the

_density

of

S- exists and

_equals

As the

_proofs

for the existence of

_6(S+)

and

_{6(S~ )}

are

similar,

we will

prove the existence of the latter and leave the easier case, the existence of

6(S+ ) ,

to the reader.

We assume that a and 1ii are

multiplicatively independent.

We are

interested in the

_growth

rate for z - oo of

in

_comparison

with

_1r(x),

the number of

_primes

up to x. The

key

idea for

computing

this

_quantity

is the

_following.

_By

formula

₍₁₀₎

and

_{corollary 8b,}

for

_{large 1}

the

_density

of

_S~

is close

_{to 2.}

In other

_words,

we

_get

a

good

approximation

of

_{S- (x)}

_{by ignoring}

the

_primes

_{up to x} that are not in

Si

for some

_’large’

_y. To make this

_precise,

we define for _{x, y} E

R&#x3E;o

the numbers

and,

for z E R U

_{oo}

with z &#x3E; _y,

For

_{each y}

E

_R&#x3E;o,

we have the

following

inequalities:

We will _prove that with _y

_{= 7}

_logx,

the limit 8- =

limx-4oo

S-(x, y) /,7r (x)

exists and that the error term

_M-(x,

_y,

_oo)

is for x -+ oo. With

the

_inequalities

₍₁₄₎

this shows that the

_density

of S- exists and

_equals

8-. First we focus on

S-(x,

y).

_By

the

_principle

of inclusion and exclusion we find

where

_P(y)

is the

_product

of all

_primes

_up to _y, and

7r~ (z, n)

= with

p inert in and

p ~

Si

for all

primes i

I nl

= with

p inert in and p

splits completely

in

Ln/Kl.

The last

_equality

follows from

_proposition

5b. To estimate 7r-

_{(x, n),}

we

need an effective version of Chebotarev’s

_density

theorem. The known

versions of such a theorem all have error terms in their statements which

are too

_large

for our _purpose.

_However,

we have the

following

conditional

Theorem 15. Let

_F/Q

be a normal extension with Galois _group

G,

let C

be a union

of conjugacy

classes

of G,

and denote the absolute value

of

the

discrarrainant

_of

F

_by

_dF.

_Define

_7rc(x)

as

_follows

7rc (x) = #~P ~

x with _p

unramified

in

F/Q

and

(p,

F/Q)

C

_Cl,

where

_(p,

denotes the Frobenius class

_of

p in

F/Q.

If

we assume the

generalized

Riemann

_hypothesis,

then there exists an absolute constant r~4

such that

_for

x &#x3E; 2 the

following inequality

holds:

with

_Li(x)

=

lo g tt

the

_logarithmic

_integral.

To

_approximate

7r-

_{(x, n),}

we

apply

theorem 15 with F =

Ln

and

If we assume GRH and use

_proposition

13 for the estimate of the

discrimi-nant of

_Ln,

we find

where

_c(n)

is the

_cardinality

of

_C(n),

and the

_implied

constant

_{only depends}

on a. If we substitute

(16)

into

(15)

with _y =

i7

log x,

we find for z - oo

the

_following:

Here we use the notation

log 2 x

for To derive formula

(17),

we used

the trivial bound

_{c(n) [Ln : Q]}

_2n3,

and the

_inequality

_P(y)

e2y.

The constant x

_{= i7}

is chosen in such a _way that the sum over n

log

x)

of the error term in

₍₁₆₎

is of order

_o(x(z))

for x -* oo.

By corollary

11,

the limit for x -~ oo of the sum in

(17)

_converges.

To finish the

_proof

of theorem 14 in the inert case, we are left with

proving

that

_M-(x,

_i7logx,oo)

=

o(7r(x»

for _{x -+ oo.} As in

Hooley’s

proof,

we use the

_following

trivial

_identity: