Chebyshev's method for number fields

J

OSÉ

F

ELIPE

V

OLOCH

Chebyshev’s method for number fields

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{1 (2000),}

p. 81-85

<http://www.numdam.org/item?id=JTNB_2000__12_1_81_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

Chebyshev’s

method

for

number

fields

par

JOSÉ

FELIPE VOLOCH

RÉSUMÉ. Nous donnons une _preuve élémentaire d’une

minora-tion

_explicite

du nombre de nombres

_premiers

_qui se

_décomposent

complètement

dans un _corps de nombres. La _{preuve qui} utilise

les

_propriétés

des coefficients binomiaux

_{s’apparente}

à

_l’approche

classique

des théorèmes de

_Chebyshev.

ABSTRACT. We

_give

an

elementary proof

of an

explicit

estimate

for the number of _primes

_splitting

_completely

in an extension of

the rationals. The

_proof

uses binomial coefficents and extends

Chebyshev’s

classical

_approach.

Chebyshev

[C]

proved

that the number of

_primes

_up to x was between

multiples

of

_x/

_log

~. In the

_simplified

form of

Chebyshev’s

method worked

out

_by

Erd6s this can be obtained

by looking

at the

_prime

factorization of the binomial coefficients

_{2,n .}

Note that

_(2n)

=

(-4)’n

(-1/2)

is the n-th

coefficient of the

_{Taylor expansion}

of

_{(1 -}

_4x)-1~2.

In the course of their

work on Grothendieck’s

_conjecture

on differential

equations,

by

considering

Pad6

_{approximations}

to

₍₁

+ =

1, 2, ...

where a is an irrational

algebraic integer,

D. and G.

_{Chudnovsky proved}

that there are

_infinitely

many

primes

which do not

split

in

Q (a) (which

is of course a

special

case of

Chebotarev’s

_density

_theorem).

The

_proof

_requires

_{estimating complicated}

expressions

in various binomial coefficients

_(ia:j).

In this paper we show

that,

at least for

_{Q(a) /Q}

_Galois,

we can _very

easily

obtain not

only

that

there are

_infinitely

_many

_primes

which

_split

_completely

in

Q((a),

but that

there are at least a

multiple

of xllll log x

(d

=

[Q(a) : Q])

such

primes

_up

to x,

by simple

estimates of

_(~).

We will also

_study

the

_prime

factorization

of

_~~~

in the

_light

of the

_present

_knowledge

about the distribution of

_primes

to

_analyse

the scope of this method.

Lemma 1. Let a E

_{C, a ~}

_N,

then

_log

_(fl)

₌

_o(n).

Proof.

This is of course

_elementary

and well-known. The function

(1

+

_x)a

is

_holomorphic

on the unit disk and has a

_singularity

at x = -1 so its

Taylor

expansion

about zero has radius of _convergence

1,

hence the result. D

For our _purposes, Lemma 1 will

_suffice,

but it is worth

_pointing

out that the estimate can be

_{substantially improved}

(see [VV]).

For now on let a be an irrational

_{algebraic integer}

such that

Q(a)/Q

is

Galois and denote

_by

a = _{al, ... , ad} the

_{conjugates of a,}

_so d

= (Q(a) :

Q~.

Put

_{f (x) _}

_ai),

the minimal

_polynomial

of a over

Q.

Under our

assumptions,

E

_{Z [z].}

Let

_An

be the absolute norm of

(~),

which is a non-zero rational number. We denote

_by

_{vp the}

_p-adic

valuation associated

to the

_prime

_p. We will often use

that,

since

Q((a)/Q

is

Galois,

for a

prime

p of

Q,

p

splits completely

in

Q(a)

if and

only

if there is a

prime

of

Q(a)

above _p which is

_split

over

Q.

Lemma 2. Assume

_{that p}

does not divide the discriminant Then

0,

if p splits completely

in

_Q(a)

and

_vp(An)

0 otherwise.

Proof. If P splits completely

in

_Q(a)

then

_Q(a)

embeds in

_Qp

and since

Q(a)/Q

is

_Galois,

all

_conjugates

of a will be in

_Qp.

Therefore, they

will

actually

be in

_Zp,

since

_they

are

_algebraic

_{integers. Now,}

for x E

_Zp

then

(2:) E

Zp,

1, ... ,

d, if p splits, giving

the first statement

of the lemma. For the second statement we note that there is no root of the minimal

_polynomial

of a in

Z/pZ

in that case for that would force the

existence of a root of

_{f (x)}

in

_Zp,

_by

Hensel’s

_lemma,

which would force p to

_split,

so _p does not divide the numerator of

_An.

D Lemma 3. Assume _p does not

_split

in

_Q(a).

Then

_vp(An)

-cn +

0(log ~,),

for

some c &#x3E; 0

_depending

on _p.

Proof.

If vp

denotes an extension

of the p-adic

valuation to

_{Zp [a]}

_then,

since

p does not

split,

Zp(a~

is

strictly bigger

than

_Zp

so there exists an

_integer

s &#x3E; 0 such that

_{vp (aj -}

_k)

_{s, j}

=

1, ... ,

d, k

_E Z. Given

j,1

j d,

there are at most

_n/p

+

₀₍₁₎

_integers

_{k, 0}

k n with

_{vp (aj -}

_k)

&#x3E; 0.

At most

n/p2

+

₀₍₁₎

of those

_satisfy

_k)

&#x3E; 1 and so _on, until at

most

_nips

+

₀₍₁₎

of those

_satisfy

_{vp(aj -}

_k)

&#x3E; s - 1 but none

satisfy

k)

&#x3E; s. As

-- -a - .

Lemma 4.

_{If p splits}

in

_Q(a)

then

_{vp (An)}

«

_log

_n/

_{log p.}

Proof.

Consider

An, a

as

_{p-adic integers.}

We have that

Given

_{j, 1 j}

_d,

there are

nip

+

₀₍₁₎

_{integers k, 0 k}

n with

However,

k)

nd,

so r «

_{log n/ log p.}

Therefore the

_p-adic

valuation of the numerator of the above

_expression

for

_An

is at

most

_{dn/ (p -}

₁₎

+

_{O(log n/ log p).}

But the last

_expression

is also a lower

bound for the

_p-adic

valuation of the denominator of the above

_expression

for

_{An, namely}

n!d.

The lemma follows. D

Theorem.

_{If S}

is the set

_of

_primes

_splitting

in

_Q(a)

then

Proof.

For x

_{suf&#x26;ciently large,}

let _y be the

_{unique positive}

solution to

f (y)

= x and

_{put n}

=

[y].

By

lemma

_1,

_log

)An )

=

o(n).

On the other

hand,

- , . , - I.. . , - - I.. . ,- -- - -

1..,,-with

_{p E S,}

then

_{p divides}

_{f (k)}

for some

_k,

0 k _n, so

_p

_{f (n) x}

and

by

lemma

_4,

_vp(An)

_{log p}

C

_log

x, so

By

lemma

_2,

for all but

_finitely

_many

_primes

_{p not} in

_S,

0 and for

_{any p V}

_S,

_vp(A,,)

-cn

_eventually.

_Therefore,

provided

there

exists at least one

_prime

p ~

S, -

_{logp »}

n W and the result follows. If S is the set of all

_primes

then the theorem follows from

Chebyshev’s original

argument.

D

It is

_possible

to estimate the

_implicit

constant in the statement of the

the-orem

by making

explicit

the estimates in Lemmas

_{1, 3}

and 4. For Lemma

1,

this is done in

_[VV]

and the estimate will

_depend

on the archimedian

ab-solute values of _a, and for suitable _a, these can be estimated in terms of the discriminant of

_GZ(a).

The constant in Lemma 4 is

_easily

estimated

and

_{depends only}

on d. The constant in Lemma does

not

_ramify.

This will make the constant in the Theorem

_depend

on the

size of the smallest

_non-split

_prime,

which in

_general

can be bounded

by

a

power of the discriminant of

Q(a)

(see

[VV]).

The

implicit

constant in the Theorem will then be a

_negative

_power of the discriminant of

Q ( a)

for x

sufficiently large.

For

_cyclotomic

fields this can be much

improved.

If a is

a

primitive

m-th root of

unity then p splits

if and

only if p -

1(modm),

so in

particular,

none of the

_{primes p}

m

_split

and the constant of the theorem in this case can be taken to be which is about

log

m, for m

_large.

Let’s

_stop

_pretending

we don’t know

anything

about the distribution

of

_primes.

The estimate

_{#fp ~ ~}

_I

is

_equivalent

to

the

_prime

ideal theorem for

_Q(a).

We will now discuss the limitations

of the above method. The contribution of a

_prime

_p that doesn’t

_split

or

so the

logarithm

of the denominator of

An

is

and this is the same as

0 (n).

_Using

lemmas 1 and 4 we

_get

that,

for any c &#x3E;

1,

O(~).

Therefore

_{large primes}

must be

_contributing

to the numerator of

_An.

Note that

_vp(An)

&#x3E; 0 for

p &#x3E; n if and

_only

if there exists

_{a, 1}

a

n, f (a) - 0(modp).

Thus, large

primes

p which contribute to the numerator of

_An

are those for which there

is a small solution to

_{f (x) - 0(modp).}

In the case that a is

_imaginary

quadratic,

Duke et al.

_[DFI]

have shown that the values of

_a/p,

where 0 a _p,

f (a)

=

0(mod p),

are

_uniformly

distributed in

_~0,1~

as _p varies.

No such result is known for

_general

_a, but a weak statement about small values of

_a/p

can be deduced from the above

_and,

_replacing

a

_by

2a we

_get

values of

_a/p

near

1/2

but it is unclear how far this can be

pushed.

Here is a numerical

_{example. Let,}

as

usual, i2

= -1 and let A be the

norm of

~50~

(A

=

A50

in the above

notation).

Its value is

approximately

A = 0.001441.... The _numerator of A is

which factors as

and we can see that indeed

_primes

much

_larger

than 50 occur. The

denom-inator of A is

which factors as

2 69344716118194234312 432472.

There is a different _way to extend

_{Chebyshev’s}

method for number

_fields,

first considered

_by

Poincax6

_[P]

for

_quadratic

fields and

_by

Landau

_[L],

in

general.

It consists of

_taking

_{generalization}

of the

_factorial,

where I runs

_through

the non-zero ideals of the number

field,

NI is the norm of I and X is a

_parameter.

This

appoach

has been combined with

deep

modern estimates in

_prime

number

_{theory by}

Friedlander

_[F]

to

_give

good

estimates for the number of

_{split primes}

_uniformly

in terms of the discriminant of the number field.

Acknowledgements:

The author would like to thank J. Vaaler and F.

_{Rodriguez Villegas}

for

_helpful

conversations and H. Lenstra for

_pointing

out an

innacuracy

in an earlier version of the _paper. We also

acknowledge

the use of the software PARI for the numerical calculations. The author

would also like to thank the TARP

_{(grant #ARP-006)}

and the the NSA

(grant MDA904-97-1-0037)

for financial

_support.

REFERENCES

[C] P.L. Chebyshev, Memoire sur les nombres premiers. J. Math. pures et appl. 17 _(1852), 366-390.

[CC] D. Chudnovsky, G. _{Chudnovsky, Applications of} Padé approximations to the Grothendieck

conjecture on linear differential equations. In Number _{theory (New} York, 1983-84),

52-100, Lecture Notes in Math. _1135, Springer, Berlin-New York, 1985.

[DFI] W. Duke, J. B. Friedlander, H. Iwaniec, Equidistribution of roots _of a _{quadratic congruence}

to _{prime moduli,} Ann. of Math. 141 _(1995), 423-441.

[F] J.B. Friedlander, Estimates _for Prime Ideals. J. Number _Theory, 12 _(1980), 101-105.

[L] E. Landau, Über die zu einem algebraischen Zahlkörper gehörige Zetafunktion und die

Ausdehnung der Tschebyscheffschen Primzahlentheorie auf das Problem der Verteilung

der Primideale. Crelle 125 _(1903), 64-188.

[P] H. _Poincaré, Extension aux nombres _{premiers complexes} des Théorèmes de M.

Tchebich-eff. J. Math. Pures _{Appl. (4)} 8 _(1892), 25-68.

[VV] J. Vaaler, J.F. Voloch, The least _{nonsplit prime} in Galois extensions of Q. J. Number

Theory, to appear.

JosA Felipe VOLOCH

Dept. of Mathematics,

Univ. of Texas, Austin, TX _78712, USA