Note on the jacobi sum $J(\chi ,\chi )$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

S

TANISLAV

J

AKUBEC

Note on the jacobi sum J(χ, χ)

Journal de Théorie des Nombres de Bordeaux, tome

7, n

o

_{2 (1995),}

p. 461-471

<http://www.numdam.org/item?id=JTNB_1995__7_2_461_0>

© Université Bordeaux 1, 1995, tous droits réservés.

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

Note

on

the

Jacobi

sum

_J(~,~).

par STANISLAV JAKUBEC

Notation

X - the Dirichlet character modulo

- Jacobi sum

- Gaussian _sum

Introduction

be the Jacobi sum,

Q«(z).

It is well known that

~ =

_p, and one

easily

_proves that

The main aim of this _{paper is} to solve the

_problem:

When is

_X)

_up

to association and

_{conjugation uniquely}

determined

_by

the solution of the

equation

We

_give

a

complete

solution in cases 1

_{= 11, 19.}

On the basis of this

result,

the

_following

_question

is answered:

When is the

_prime

2 an 11-th _resp. a 19-th _power modulo _p if _{p is} not

representable by

the

_quadratic

form

X2 +

_{l ly2 ,}

resp.

x2

+

19y2.

1991 Mathematics Subject Classification. Primary 11R18.. Manuscrit requ le 19 janvier 1993

We shall now

_present

a _{survey of} results obtained

_by

_solving

the

problem

when the

_prime

2 is an 1-th _power modulo _p.

Jacobi has

_given

_{necessary and sufficient} conditions for

_{primes q}

37 to be cubes modulo

_primes

p = 1

(mod 3).

For

example,

he proves the

following

PROPOSITION 1. ~ is a cube modulo

_{p if}

and

_onty

0

_{(mod 2),}

where

Emma Lehmer

_[2]

finds the

_following

result:

PROPOSITION 2. Let p = 1

_{(mod 5)}

be a

_prime.

Then 2 is a

_fifth

_power

modulo p

if

and

only

if

x = 0

(mod 2),

where

(x,

u, v,

w)

is one

of

the

ex-actly

four

solutions

_(x,

u, v,

w), (x,

-u, -v,

w),

(x,

v, -u,

-w), (x,

-v, u,

-w)

of

the

_diophantine

_system

_(Dickson):

P.A. Leonard and K.S. Williams

_[4]

prove the

following

PROPOSITION 3. Let _p = 1

_{(mod 7)}

be a

_prime.

Then 2 is a -seventh power

modulo _p

_if

and

_only

_{if x, =-}

0

_(mod

_2),

where

_{(xl, x2, ... , z6 )}

is one

_of

the

exactly

six non-trivial solutions

_of

the

diophantine

system

of

equations

PROPOSITION 4. Let p - 1

(mod 11)

be a

_prime.

Then 2 is an eleventh

power modulo p

if

and

only if

a certain condition

involving

solutions

_of

a _very

complicated

diophantine

_system

holds

(the

exact statement may be

seen in

_[31).

J.C.

Parnami,

M.K.

_Agrawal

and A.R.

_Rajwade

_[5]

have the

_following

PROPOSITION 5. Let p = 1

_{(mod l).}

Then 2 is an l-th power rriodulo p

if

and

_{only if}

where

_(al,

a2, ... ,

ai-I)

is one

of

the

exactly

l -1 solutions

of

the

diophan-tine

_system

_of

_equations

where

_{A(n) is}

the least

_non-negative

residue

_of

n modulo

_l,

and gk is the

automorphism (I

-&#x3E;

(f,

Now let XX = _p, and let

_X)

be associated with the number

_X,

i.e.

J(x,

x)

=

_eX,

where E is _{a unit} of the field Then

implies

e"£ =1 and hence c =

_(2013~)~.

So we have

Let 2 be a

_primitive

root modulo l. Consider a residue class field

Z((1) /(2)

of the

_degree

_f

= 1 - 1 over

_Z/2Z.

_{Let g}

be a

_generator

of

the

_{multiplicative}

_group

_{(Z«(I)/(2»*}

of the field

_Z(~l)/(2)

such that there

LEMMA 1.

_Every

unit E

of

the

field

Q(~1)

is _{power in} the _group

(Z«(z)j(2»*,

m =

1;1.

Proof.

Let e =

_g’~

(mod

2).

It is necessary _{to prove} that

, in.

Consider

the unit hence 6"1 must be a root of

_1,

therefore

_ell

= 1.

Further,

It follows that

_{2Ln(2"‘ - 1) -}

0

_(mod

2l-1 -

_1),

and therefore n - 0

(mod 2ml+l). 0

LEMMA 2. Let 2 be a

_primitive

root modulo l. For a natural number _a,

0 a L - 1 the

following identity

holds

(the

decomposition

into

_binary

_system),

where r,, =- l = 2 -

ind(n)

+

_ind(a)

(mod

l-l),

0 rn

l -1,

and

ind(x)

is the index

of

the element x in the group

(Z/lZ)*

under the base

2,

i.e.

(mod l).

Proof.

The lemma can be

_{readily proved}

when the rational number

_a

_is

"0

expressed

in the

_binary

_{system, ’}

_{= E}

_{an2 -n.}

n=l

LEMMA 3. The

_{factorisation of}

the Jacobi sum

_J(X,

_X)

into

prime

divisors

1-1

11

is a

_prime

_{divisor of}

n=1 n

1

the

_field

_{Q(l), pip,}

and is an

_automorphism

=

Cn

n n

Proof. According

to

_[1],

I-I

The factorisation

₁₁

is obtained

_using

the

Consider a divisor

where ji

=

_{0; 1,}

and define

LEMMA 4. The

_{factorisation .}

is a

_{conjugation of}

the

factori-Proof.

Let A be a

_conjugation

of

J(X,

X).

Then A =

X-)

for _{some s.}

If s = k

then we can write

2.

_Conversely,

let

By

Lemma

_2,

Since the

_expansion

of a number in the

binary

_system

is

uniquely

deter-mined,

Lemma 4 is

_proved.

D

For

_pip,

denote

_{by hi,}

the least natural number such that a

_principal

divisor =

(a).

THEOREM 1. Let 2 be a

_primitive

root and tet a =

gm

_(mod

_2),

If

then X

_is,

_up to association and

_{conjugation, equal}

to the Jacobi sum

J(x, X).

Proof. According

to Lemma

1,

the choice of a

_generator

a of the

principal

divisor =

(a)

is not substantial.

Suppose

the factorisation of X into

_prime

divisors of the field

_Q«(l)

is

It is necessary to _prove that this factorisation is a

_conjugate

of the

factori-Clearly

hence

where e is a unit of the field

Q(~1).

But a = 1

implies

e =

(-(1 ) ,

and therefore

From X = 1

_(mod

₂₎

we obtain

hence

Consequently

It is easy to _prove that the condition XX = _p

_gives

From _{the congruences}

₍₁₎

and

_(2),

_using

the

_assumption

(M, 2 ~+i ) -

1

and the fact that

(2"’~ -

1, 2m

+

₁₎

=

_1,

_we obtain the congruences

hence

From XX = _{p, it} follows

1,

and this

implies

a l - 1. Due to

Lemma

_4,

Theorem 1 is

_proved.

R

Remark. If

_(M,

₊₎

= d _&#x3E;

1,

then instead of the congruence

(3)

we

_get

the congruence

It can be

_proved

that _{this congruence} has

_always

the solution

which is not

_{corresponding}

to the

_conjugates

of the Jacobi sum

J(x, x~ ~

The

_question

of whether for p with d &#x3E; 1 the Jacobi sum

J(x,

x)

can be

uniquely

determined transforms in the

_question,

for

_{which ji}

the divisor

COROLLARY 1. Let 2 be a

_primitive

root modulo

_1,

and let the class number

of

the

_field

_Q«(l)

be

_equal

to 1. Then the Jacobi sum

J(X, X)

is

uniquely

determined, up

to association and

_conjugation,

_from

the solution

_of

the

equation

XX =

p, X

=1

(mod

2) X

_E

Z(~1)

if

and

only if

(M,

=1.

Proo f.

This follows from the

preceding Remark,

because in such a _case,

every divisor is

principal.

EXAMPLE 1. 1 =

_5,

=

22t1

= 1. It

follows

that

for

all _{p one} has

d =

_1,

hence the Jacobi sum

_{J(X, X)}

is

_{uniquely determined,}

up to

asso-ciation and

_conjugation,

_by

the solution

_of

the

_equation

XX = _p, X ! 1

(mod

2),

X E

It _{is easy} to see, that if we want to answer the

_question,

for which

_primes

p is the Jacobi sum

J(X,

X)

_{uniquely determined,}

_up to association and

conjugation,

by

the solution of the

_equation

XX = _p X -1

(mod 2),

X _E

Z(CI),

then we must know for which

_primes

p the number a, where

N(a) =

p, is a third _{power in} the group

(Z(~l)/(2))*.

This

question

is solved in the

following

lemma.

LEMMA 5. Let 2 be a

_primitive

root modulo 1 =- 3

_{(mod 4), plp,}

and

_php

=

a.

Then a is a third _{power in} the _groups

(Z(~l)/(2))*

if

and

onty

=

X2

+

ly2,

_{where x, y are} not

_{simultaneously}

divisible

_bg

_p.

Proof. By

Lemma

_1,

the choice of a

_generator

a of the

principal

divisor

php

is not substantial. Consider the

_product

Let

So we have a’ = b’ = 1

_(mod

_2).

Hence

therefore

Let

_conversely

This

_{implies a’ =}

b’

= 1

_{(mod 2),}

hence

Let Then since

,B

is invariant on

where (7)

=

1,

_we set that

But if

₍₇₎

_

-1,

then

7n{p)

does not

divide!3

(in

the

opposite

case we

would

_get

hence a

contradiction).

It

_implies

therefore

and

From this we

finally

obtain

THEOREM 2. Let 1 =

11; 19,

and let _p = 1

_{(mod l),}

_4p

=

A2

+ lB2.

The

Jacobi sum

J(X, X)

is

uniquely

determined,

_{up to}

_conjugation

and

associa-tion,

by

the solution

_of

if

and

_only

_{(mod 2).}

Proof.

The

_proof

follows from Lemma 5 and

_Corollary

1. D

By Proposition

5 and Theorem

_2,

we come to the

_following

THEOREM 3. Let I =

11; 19,

and let

p - 1

(mod 1),

4p

= A2

+

_1B2,

A - B - 1

_{(mod 2).}

Then 2 is an l-th _power modulo

_{p if}

and

_{only if}

where

_(al,

a2, ... ,

al-1)

is one

of

the

_exactly

1 - 1 solutions

of

the

diophan-tine

_system

_of

_equations

Remark. As we can see, Theorem 2 enables us to remove condition

_(iii)

of

Proposition

5.

REFERENCES

[1]

H. _{HASSE, Vorlesungen} uber _{Zahlentheorie,} Berlin 1950.

[2]

E. _LEHMER, The quintic character _{of 2 and 3,} Duke math. J. 18

_(1951),

11-18.

[3]

P. A. _LEONARD, B. C. MORTIMER and K. S. _WILLIAMS, The eleventh _power

[4]

P. A. LEONARD and K. S. _WILLIAMS, The _septic character of 2,3,5 and _7, Pacific J. Math. 52

_(1974),

143-147.

[5]

J. C. _PARNAMI, M. K. AGRAWAL and A. R. _RAJWADE, Criterion for 2 to be l-th power, Acta Arithmetica 43

_(1984),

361-364.

Stanislav JAKUBEC Mathematical Insti.tute of Slovak Academy of Sciences Stefanikova 49

814 73 Bratislava Slovakia