Eisenstein reciprocity

Statement of the theorem

Let ${\displaystyle m\in \mathbb{Z}}$ be an odd prime and ${\displaystyle a\in \mathbb{Z}}$ an integer relatively prime to ${\displaystyle m.}$ Then

First supplement

${\displaystyle {\left(\frac{{\zeta }_m}{a}\right)}_m={\zeta }_m^{\frac{a^{m-1}-1}{m}}.}$

Second supplement

${\displaystyle {\left(\frac{1-{\zeta }_m}{a}\right)}_m={\left(\frac{{\zeta }_m}{a}\right)}_m^{\frac{m+1}{2}}.}$

Eisenstein reciprocity

Let ${\displaystyle \alpha \in {\mathcal{O}}_m}$ be primary (and therefore relatively prime to ${\displaystyle m}$), and assume that ${\displaystyle \alpha }$ is also relatively prime to ${\displaystyle a}$. Then

${\displaystyle {\left(\frac{\alpha }{a}\right)}_m={\left(\frac{a}{\alpha }\right)}_m.}$

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 ${\displaystyle K\supset \mathbb{Q}({\zeta }_l)}$ is an arbitrary algebraic number field containing the ${\displaystyle l}$-th roots of unity for a prime ${\displaystyle l}$, then Eisenstein's law for ${\displaystyle l}$-th powers holds in ${\displaystyle K.}$

Applications

First case of Fermat's Last Theorem

Assume that ${\displaystyle p}$ is an odd prime, that ${\displaystyle x^p+y^p+z^p=0}$ for pairwise relatively prime integers (i.e. in ${\displaystyle \mathbb{Z}}$) ${\displaystyle x,y,z}$ and that ${\displaystyle p\nmid xyz.}$

This is the first case of Fermat's Last Theorem. (The second case is when ${\displaystyle p\mid xyz.}$) Eisenstein reciprocity can be used to prove the following theorems

(Wieferich 1909) Under the above assumptions, ${\displaystyle 2^{p-1}\equiv 1(mod{p}^2).}$

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 ${\displaystyle 3^{p-1}\equiv 1(mod{p}^2).}$

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 ${\displaystyle r\mid x,r^{p-1}\equiv 1(mod{p}^2).}$

(Furtwängler 1912) Under the above assumptions, for every prime ${\displaystyle r\mid (x-y),r^{p-1}\equiv 1(mod{p}^2).}$

(Vandiver) Under the above assumptions, if in addition ${\displaystyle p>3,}$ then ${\displaystyle x^p\equiv x,y^p\equiv y}$, and ${\displaystyle z^p\equiv z(mod{p}^3).}$

Powers mod most primes

Eisenstein's law can be used to prove the following theorem (Trost, Ankeny, Rogers). Suppose ${\displaystyle a\in \mathbb{Z}}$ and that ${\displaystyle l\nmid a}$ where ${\displaystyle l}$ is an odd prime. If ${\displaystyle x^l\equiv a(modp)}$ is solvable for all but finitely many primes ${\displaystyle p}$, then ${\displaystyle a=b^l.}$