Zolotarev’s lemma

Zolotarev’s lemma

We will identify the ring

ℤ

n

of

integers

modulo

n

, with the set

{

0

,

1

,

…

⁢

n

-

1

}

.

Lemma 1

(Zolotarev).

For any

prime number

p

and
any

m

∈

Z

p

*

, the

Legendre symbol

(

m

p

)

is equal
to the

signature

(

http://planetmath.org/SignatureOfAPermutation

) of the

permutation

τ

m

:

x

↦

m

⁢

x

of

Z

p

*

.

Proof.

We write

ϵ

⁢

(

σ

)

for the signature of any permutation

σ

. If

σ

is a

circular permutation

on a set of

k

elements

, then

ϵ

⁢

(

σ

)

=

(

-

1

)

k

-

1

. Let

i

be the order of

m

in

ℤ

p

*

. Then the permutation

τ

m

consists of

(

p

-

1

)

/

i

orbits, each of size

i

, whence

ϵ

⁢

(

τ

m

)

=

(

-

1

)

(

i

-

1

)

⁢

(

p

-

1

)

/

i

If

i

is even, then

m

(

p

-

1

)

/

2

=

m

i

2

⁢

p

-

1

i

=

(

-

1

)

p

-

1

i

=

ϵ

⁢

(

τ

m

)

And if

i

is odd, then

2

⁢

i

divides

p

-

1

, so

m

(

p

-

1

)

/

2

=

m

i

⁢

p

-

1

2

⁢

i

=

1

=

ϵ

⁢

(

τ

m

)

.

In both cases, the lemma follows from Euler’s criterion.
∎

Lemma 1 extends easily from the Legendre symbol to the

Jacobi symbol

(

m

n

)

for odd

n

. The following is Zolotarev’s penetrating proof of the quadratic reciprocity law, using Lemma 1.

Lemma 2

.

Let

λ

be the permutation of the set

A

m

⁢

n

=

{

0

,

1

,

…

,

m

-

1

}

×

{

0

,

1

,

…

,

n

-

1

}

which maps the

k

th element of the

sequence

(

0

,

0

)

⁢

(

0

,

1

)

⁢

…

⁢

(

0

,

n

-

1

)

⁢

(

1

,

0

)

⁢

…

⁢

(

1

,

n

-

1

)

⁢

(

2

,

0

)

⁢

…

⁢

(

m

-

1

,

n

-

1

)

,

to the

k

th element of the sequence

(

0

,

0

)

⁢

(

1

,

0

)

⁢

…

⁢

(

m

-

1

,

0

)

⁢

(

0

,

1

)

⁢

…

⁢

(

m

-

1

,

1

)

⁢

(

0

,

2

)

⁢

…

⁢

(

m

-

1

,

n

-

1

)

,

for every

k

from

1

to

m

⁢

n

. Then

ϵ

⁢

(

λ

)

=

(

-

1

)

m

⁢

(

m

-

1

)

⁢

n

⁢

(

n

-

1

)

/

4

and if

m

and

n

are both odd,

ϵ

⁢

(

λ

)

=

(

-

1

)

(

m

-

1

)

⁢

(

n

-

1

)

/

4

.

Proof.

We will use the fact that the signature of a permutation of
a finite

totally ordered set

is determined by the number of
inversions of that permutation.
The sequence

(

0

,

0

)

,

(

0

,

1

)

⁢

…

defines on

A

m

⁢

n

a

total order

≤

in which the

relation

(

i

,

j

)

<

(

i

′

,

j

′

)

means

i

<

i

′

or

(

i

=

i

′

and

j

<

j

′

)

.

But

λ

⁢

(

i

′

,

j

′

)

<

λ

⁢

(

i

,

j

)

means

j

′

<

j

or

(

j

′

=

j

and

i

′

<

i

)

.

The only pairs

(

(

i

,

j

)

,

(

i

′

,

j

′

)

)

that get inverted are, therefore,
the ones with

i

<

i

′

and

j

>

j

′

.
There are indeed

(

m

2

)

⁢

(

n

2

)

such pairs, proving the first

formula

, and the second follows easily.
∎

And finally, we proceed to prove

quadratic reciprocity

. Let

p

and

q

be distinct

odd primes

. Denote by

π

the
canonical ring

isomorphism

ℤ

p

⁢

q

→

ℤ

p

×

ℤ

q

.
Define two permutations

α

and

β

of

ℤ

p

×

ℤ

q

by

α

⁢

(

x

,

y

)

=

(

q

⁢

x

+

y

,

y

)

and

β

⁢

(

x

,

y

)

=

(

x

,

x

+

p

⁢

y

)

.

Finally, define a map

λ

:

ℤ

p

⁢

q

→

ℤ

p

⁢

q

by

λ

⁢

(

x

+

q

⁢

y

)

=

p

⁢

x

+

y

for

x

∈

{

0

,

1

,

…

⁢

q

-

1

}

and

y

∈

{

0

,

1

,

…

⁢

p

-

1

}

.
Evidently

λ

is a permutation.

Note that we have

π

⁢

(

q

⁢

x

+

y

)

=

(

q

⁢

x

+

y

,

y

)

and

π

⁢

(

x

+

p

⁢

y

)

=

(

x

,

x

+

p

⁢

y

)

, so
therefore

π

∘

λ

∘

π

-

1

∘

α

=

β

.

Let us compare the signatures of the two sides. The permutation

m

↦

q

⁢

x

+

y

is the

composition

of

m

↦

q

⁢

x

and

m

↦

m

+

y

. The latter has signature

1

, whence by Lemma 1,

ϵ

⁢

(

α

)

=

(

q

p

)

q

=

(

q

p

)

and similarly

ϵ

⁢

(

β

)

=

(

p

q

)

p

=

(

p

q

)

.

By Lemma 2,

ϵ

⁢

(

π

∘

λ

∘

π

-

1

)

=

(

-

1

)

(

p

-

1

)

⁢

(

q

-

1

)

/

4

.

Thus

(

-

1

)

(

p

-

1

)

⁢

(

q

-

1

)

/

4

⁢

(

q

p

)

=

(

p

q

)

which is the quadratic reciprocity law.

Reference

G. Zolotarev, Nouvelle démonstration de la loi de réciprocité de Legendre,
Nouv. Ann. Math (2), 11 (1872), 354-362

Title

Zolotarev’s lemma

Canonical name

ZolotarevsLemma

Date of creation

2013-03-22 13:28:25

Last modified on

2013-03-22 13:28:25

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

15

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11A15

Related topic

GaussLemma

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