Eisenstein reciprocity

Background and notation

Let   m > 1 be an integer, and let   O m   be the ring of integers of the m-th cyclotomic field   Q ( ζ m ) ,   where   ζ m = e 2 π i 1 m   is a primitive m-th root of unity.

The numbers ζ m , ζ m 2 , … ζ m m = 1 are units in O m . (There are other units as well.)

Primary numbers

A number α ∈ O m is called primary if it is not a unit, is relatively prime to m , and is congruent to a rational (i.e. in Z ) integer ( mod ( 1 − ζ m ) 2 ) .

The following lemma shows that primary numbers in O m are analogous to positive integers in Z .

Suppose that α , β ∈ O m and that both α and β are relatively prime to m . Then

There is an integer c making ζ m c α primary. This integer is unique ( mod m ) .

if α and β are primary then α ± β is primary, provided that α ± β is coprime with m .

if α and β are primary then α β is primary.

α m is primary.

The significance of   1 − ζ m  which appears in the definition is most easily seen when   m = l   is a prime.  In that case   l = ( 1 − ζ l ) ( 1 − ζ l 2 ) … ( 1 − ζ l l − 1 ) .   Furthermore, the prime ideal   ( l )   of   Z   is totally ramified in  Q ( ζ l )

( l ) = ( 1 − ζ l ) l − 1 ,

  and the ideal   ( 1 − ζ l )   is prime of degree 1.

m-th power residue symbol

For α , β ∈ O m , the m-th power residue symbol for O m is either zero or an m-th root of unity:

( α β ) m = { ζ  where  ζ m = 1  if  α  and  β  are relatively prime 0  otherwise .

It is the m-th power version of the classical (quadratic, m = 2) Jacobi symbol (assuming α and β are relatively prime):

If η ∈ O m and α ≡ η m ( mod β ) then ( α β ) m = 1.

If ( α β ) m ≠ 1 then α is not an m-th power ( mod β ) .

If ( α β ) m = 1 then α may or may not be an m-th power ( mod β ) .

Statement of the theorem

Let   m ∈ Z   be an odd prime and   a ∈ Z   an integer relatively prime to   m .   Then

First supplement

( ζ m a ) m = ζ m a m − 1 − 1 m .  

Second supplement

( 1 − ζ m a ) m = ( ζ m a ) m m − 1 2 .  

Eisenstein reciprocity

Let   α ∈ O m be primary (and therefore relatively prime to   m ), and assume that   α   is also relatively prime to   a   Then

( α a ) m = ( a α ) m .  

Proof

The theorem is a consequence of the Stickelberger relation.

Weil (1975) gives a historical discussion of some early reciprocity laws, including a proof of Eisenstein's law using Gauss and Jacobi sums that is based on Eisenstein's original proof.

Generalization

In 1922 Takagi proved that if K ⊃ Q ( ζ l )   is an arbitrary algebraic number field containing the l -th roots of unity for a prime l , then Eisenstein's law for l -th powers holds in K .

First case of Fermat's last theorem

Assume that p is an odd prime, that x p + y p + z p = 0   for pairwise relatively prime integers (i.e. in Z )   x , y , z and that p ∤ x y z .

This is the first case of Fermat's last theorem. (The second case is when p ∣ x y z . )   Eisenstein reciprocity can be used to prove the following theorems

(Wieferich 1909) Under the above assumptions,   2 p − 1 ≡ 1 ( mod p 2 ) .

The only primes below 6.7×10¹⁵ that satisfy this are 1093 and 3511. See Wieferich primes for details and current records.

(Mirimanoff 1911) Under the above assumptions   3 p − 1 ≡ 1 ( mod p 2 ) .

Analogous results are true for all primes ≤ 113, but the proof does not use Eisenstein's law. See Wieferich prime#Connection with Fermat's last theorem.

(Furtwängler 1912) Under the above assumptions, for every prime   r ∣ x , r p − 1 ≡ 1 ( mod p 2 ) .

(Furtwängler 1912) Under the above assumptions, for every prime   r ∣ ( x − y ) , r p − 1 ≡ 1 ( mod p 2 ) .

(Vandiver) Under the above assumptions, if in addition   p > 3 ,   then   x p ≡ x , y p ≡ y   and   z p ≡ z ( mod p 3 ) .

Powers mod most primes

Eisenstein's law can be used to prove the following theorem (Trost, Ankeny, Rogers).   Suppose   a ∈ Z   and that   l ∤ a   where   l   is an odd prime. If   x l ≡ a ( mod p )   is solvable for all but finitely many primes   p   then   a = b l .