User:Virginia-American/Sandbox/Eisenstein reciprocity

Eisenstein's reciprocity law is a theorem in algebraic number theory first proven by Gotthold Eisenstein in 1850.

Reciprocity laws are a collection of theorems in number theory. The name "reciprocity" (coined by Legendre) refers to the fact that they state conditions under whcich the congruence xn ≡ p (mod q) has a solution in terms of the solvability of xn ≡ q (mod p). Ireland and Rosen say The Eisenstein reciprocity law generalizes some of our previous work on quadratic and cubic reciprocity. It lies midway between these special cases and the more general reciprocity laws investigated by Kummer and Hilbert, proven first by Furtwängler and then in full generality by Artin and Hasse.

Lemmermeyer begins the chapter on Eisenstein reciprocity In order to prove higher reciprocity laws, the methods known to Gauss were soon found to be inadequate. The most obvious obstacle, namely the fact that the unique factorization theorem fails to hold for the rings $$\mathbb{Z}[\zeta_l],$$ was overcome by Kummer through the invention of his ideal numbers. The direct generalization of the proofs for cubic and quartic reciprocity, however, did not yield the general reciprocity theorem for $$l$$-th powers; indeed, the most general reciprocity law that could be proved within the cyclotomic framework is Eisenstein's reciprocity law. ...

Although Eisenstein's reciprocity law is only a very special case of more general reciprocity laws, it turned out to be an indispensable step for proving these general laws until Furtwängler succeeded in finally giving a proof of the reciprocity law in $$\mathbb{Q}(\zeta_l)$$ without the help of Eisenstein's reciprocity law. It should be also noted that Eisenstein's reciprocity law holds for all primes $$l$$, whereas Kummer had to assume that $$l$$ is regular, i.e. that $$l$$ does not divide the class number of $$\mathbb{Q}(\zeta_l).$$

Background and notation
Let $$m > 1$$ be an integer, and let   $$\mathcal{O}_m$$  be the ring of integers of the m-th cyclotomic field   $$\mathbb{Q}(\zeta_m),$$  where  $$\zeta_m=e^{2 \pi i\frac{1}{m}}$$  is a   primitive m-th root of unity.

Primary numbers
A number $$\alpha\in\mathcal{O}_m$$ is called primary if it is not a unit, is relatively prime to $$m$$, and is congruent to a rational (i.e. in $$\mathbb{Z}$$) integer $$\pmod{(1-\zeta_m)^2}.$$

m-th power residue symbol
For $$\alpha, \beta\in\mathcal{O}_m,$$ the m-th power residue symbol for $$\mathcal{O}_m $$ is either zero or an m-th root of unity:



\left(\frac{\alpha}{\beta }\right)_m = \begin{cases} \zeta \mbox{ where }\zeta^m=1&\mbox{ if }\alpha\mbox{ and }\beta\mbox{ are relatively prime}\\ 0 &\mbox{ otherwise}\\ \end{cases} $$ It is the m-th power version of the classical (quadratic, m = 2) Jacobi symbol:

\mbox{If } \eta\in\mathcal{O}_m \mbox{ and } \alpha\equiv\eta^m\pmod{\beta} \mbox{ then } \left(\frac{\alpha}{\beta }\right)_m = 1. $$



\mbox{If } \left(\frac{\alpha}{\beta }\right)_m \neq1 \mbox{ then }\alpha \mbox{ is not an }m\mbox{-th power}\pmod{\beta}. $$



\mbox{If } \left(\frac{\alpha}{\beta }\right)_m =1 \mbox{ then }\alpha \mbox{ may or may not be an }m\mbox{-th power}\pmod{\beta}. $$

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

First supplement


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

Second supplement


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

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



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

Proof
The theorem is a consequence of the Stickelberger relation.

Generalization
In 1922 Takagi proved that if $$K\supset\mathbb{Q}(\zeta_l)$$ is an arbitrary algebraic number field containing the $$l$$-th roots of unity for a prime $$l$$, then Eisenstein's law holds in $$K.$$

First case of Fermat's last theorem
Eisenstein reciprocity is used in some proofs of Wieferich's, Mirimanoff's and Furtwängler's theorems. These four exercises are from Lemmermeyer:

I. (Furtwängler 1912) Let $$p$$ be an odd prime, and assume that $$x^p+y^p+z^p=0\;$$ for pairwise relatively prime integers $$x,y,z\in\mathbb{Z}$$ with $$p\nmid xyz.\;\;$$ Use the unique factorization theorrem for prime ideals to deduce that $$(x+y\zeta^i) = \mathfrak{A}_i^p\;$$ for ideals $$\mathfrak{A}_i,\;\;i=0,1,\dots,p-1.\;\;$$. Show that $$\alpha=\zeta^yx+\zeta^{-x}y$$ is semi-primary. Now use Eisenstein's reciprocity law to deduce that $$(\tfrac{\alpha}{r})_p=(\tfrac{r}{\alpha})_p=(\tfrac{r}{\mathfrak{A}_j})_p^p\;\;$$ for each prime $$r\mid x$$ and deduce that $$r^{p-1}\equiv1\pmod{p^2}.$$

'''II. (Wieferich 1909)''' Suppose $$x^p+y^p+z^p=0\;$$ for some odd prime $$p\nmid xyz;\;\;$$ then $$2^{p-1}\equiv 1\pmod{p^2}.\;\;$$ (Hint: Use the preceding exercise)

Remark. Primes $$p\;$$ satisfying $$2^{p-1}\equiv 1\pmod{p^2}\;\;$$ are called Wieferich primes. The only Wieferich primes below 4×1012 are 1093 and 3511.

'''III. (Furtwängler 1912)''' Let $$p$$ be an odd prime, and assume that $$x^p+y^p+z^p=0\;$$ for pairwise relatively prime integers $$x,y,z\in\mathbb{Z}$$ with $$p\nmid xyz.\;\;$$ Assume moreover that $$p\nmid(x^2-y^2)$$ Then $$r^{p-1}\equiv 1\pmod{p^2}$$ for every prime $$r\mid(x-y).$$

'''IV. (Mirimanoff 1911)''' Suppose $$p>3$$ is prime, $$p\nmid xyz,\;\;$$ and $$x^p+y^p+z^p=0.\;$$ Then $$3^{p-1}\equiv 1\pmod{p^2}.$$

Powers mod most primes
Eisenstein's law can be used to prove

Theorem (Trost, Ankeny, Rogers). Suppose $$a\in\mathbb{Z}$$ and that $$l\nmid a$$ where $$l$$ is an odd prime. If $$x^l\equiv a\pmod{p}$$ is solvable for all but finitely many primes $$p$$ then $$a=b^l.$$