cubic reciprocity law

cubic reciprocity law

In a ring

ℤ

/

n

⁢

ℤ

, a

cubic residue

is just a value of the

function

x

3

for some

invertible element

x

of the ring. Cubic residues display
a reciprocity phenomenon

similar

to that seen with

quadratic
residues

. But we need some preparation in order to state the cubic
reciprocity law.

ω

will denote

-

1

+

i

⁢

3

2

, which is one of the

complex

cube roots

of

1

.

K

will denote the ring

K

=

ℤ

⁢

[

ω

]

. The elements of

K

are the

complex numbers

a

+

b

⁢

ω

where

a

and

b

are

integers

. We define the norm

N

:

K

→

ℤ

by

N

⁢

(

a

+

b

⁢

ω

)

=

a

2

-

a

⁢

b

+

b

2

or equivalently

N

⁢

(

z

)

=

z

⁢

z

¯

.

Whereas

ℤ

has only two units (meaning invertible elements), namely

±

1

,

K

has six, namely all the sixth roots of 1:

±

1

±

ω

±

ω

2

and we know

ω

2

=

-

1

-

ω

. Two nonzero

elements

α

and

β

of

K

are said
to be

associates

if

α

=

β

⁢

μ

for some unit

μ

. This
is an

equivalence relation

, and any nonzero element has six associates.

K

is a

principal ring

, hence has

unique factorization

. Let us call

ρ

∈

K

“

irreducible

” if the condition

ρ

=

α

⁢

β

implies
that

α

or

β

, but not both, is a unit.
It turns out that the

irreducible elements

of

K

are (up to

multiplication

by units):

– the number

1

-

ω

, which has norm 3. We will denote it by

π

.

–

positive

real integers

q

≡

2

(

mod

3

)

which are prime in

ℤ

.
Such integers are called

rational primes

in

K

.

– complex numbers

q

=

a

+

b

⁢

ω

where

N

⁢

(

q

)

is a prime in

Z

and

N

⁢

(

q

)

≡

1

(

mod

3

)

.

For example,

3

+

2

⁢

ω

is a prime in

K

because its norm, 7, is
prime in

ℤ

and is 1 mod 3; but 7 is not a prime in

K

.

Now we need some convention whereby at most one of any six associates
is called a prime. By convention, the following numbers are nominated:

– the number

π

.

– rational primes (rather than their negative or complex associates).

– complex numbers

q

=

a

+

b

⁢

ω

where

N

⁢

(

q

)

≡

1

(

mod

3

)

is
prime in

ℤ

and

a

≡

2

(

mod

3

)

b

≡

0

(

mod

3

)

.

One can verify that this selection exists and is unambigous.

Next, we seek a three-valued function analogous to the
two-valued quadratic residue

character

x

↦

(

x

p

)

.
Let

ρ

be a prime in

K

, with

ρ

≠

π

. If

α

is any
element of

K

such that

ρ

∤

α

, then

α

N

⁢

(

ρ

)

-

1

≡

1

(

mod

ρ

)

.

Since

N

⁢

(

ρ

)

-

1

is a

multiple

of 3, we can define a function

χ

ρ

:

K

→

{

1

,

ω

,

ω

2

}

by

χ

ρ

⁢

(

α

)

≡

α

(

N

⁢

(

ρ

)

-

1

)

/

3

⁢

if

⁢

ρ

∤

α

χ

ρ

⁢

(

α

)

=

0

if

ρ

∣

α

.

χ

ρ

is a character, called the cubic residue character mod

ρ

.
We have

χ

ρ

⁢

(

α

)

=

1

if and only if

α

is a nonzero cube
mod

ρ

. (Compare Euler’s criterion.)

At last we can state this famous result of Eisenstein and Jacobi:

Theorem

(Cubic Reciprocity Law):

If

ρ

and

σ

are any two distinct
primes in

K

, neither of them

π

, then

χ

ρ

⁢

(

σ

)

=

χ

σ

⁢

(

ρ

)

.

The quadratic reciprocity law has two “supplements” which describe

(

-

1

p

)

and

(

2

p

)

. Likewise the cubic law has this supplement,
due to Eisenstein:

Theorem:

For any prime

ρ

in

K

, other than

π

,

χ

ρ

⁢

(

π

)

=

ω

2

⁢

m

where

m

=

(

ρ

+

1

)

/

3

if

ρ

is a rational prime

m

=

(

a

+

1

)

/

3

if

ρ

=

a

+

b

⁢

ω

is a complex prime.

Remarks:

Some writers refer to our “irreducible” elements
as “primes” in

K

; what we have called primes, they call “

primary

primes”.

The quadratic reciprocity law would take a simpler form if we were to
make a different convention on what is a prime in

ℤ

, a convention
similar to the one in

K

: a prime in

ℤ

is either 2 or an
irreducible element

x

of

ℤ

such that

x

≡

1

(

mod

4

)

.
The primes would then be 2, -3, 5, -7, -11, 13, …and the QRL
would say simply

(

p

q

)

⁢

(

q

p

)

=

1

for any two distinct

odd primes

p

and

q

.

Title

cubic reciprocity law

Canonical name

CubicReciprocityLaw

Date of creation

2013-03-22 13:41:26

Last modified on

2013-03-22 13:41:26

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Topic

Classification

msc 11A15

Related topic

QuadraticReciprocityRule

Defines

cubic residue

Defines

cubic residue character

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