Eisenstein reciprocity

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In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by Eisenstein (1850), though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839.[1]

Background and notation[edit]

Let   be an integer, and let     be the ring of integers of the m-th cyclotomic field     where    is a primitive m-th root of unity.

The numbers are units in (There are other units as well.)

Primary numbers[edit]

A number is called primary[2][3] if it is not a unit, is relatively prime to , and is congruent to a rational (i.e. in ) integer

The following lemma[4][5] shows that primary numbers in are analogous to positive integers in

Suppose that and that both and are relatively prime to Then

  • There is an integer making primary. This integer is unique
  • if and are primary then is primary, provided that is coprime with .
  • if and are primary then is primary.
  • is primary.

The significance of    which appears in the definition is most easily seen when     is a prime.  In that case     Furthermore, the prime ideal     of     is totally ramified in 

  and the ideal     is prime of degree 1.[6][7]

m-th power residue symbol[edit]

For the m-th power residue symbol for is either zero or an m-th root of unity:

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

  • If and then
  • If then is not an m-th power
  • If then may or may not be an m-th power

Statement of the theorem[edit]

Let     be an odd prime and     an integer relatively prime to     Then

First supplement[edit]

 [8]

Second supplement[edit]

 [8]

Eisenstein reciprocity[edit]

Let   be primary (and therefore relatively prime to   ), and assume that    is also relatively prime to  . Then[8][9]

Proof[edit]

The theorem is a consequence of the Stickelberger relation.[10][11]

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[edit]

In 1922 Takagi proved that if   is an arbitrary algebraic number field containing the -th roots of unity for a prime , then Eisenstein's law for -th powers holds in [12]

Applications[edit]

First case of Fermat's last theorem[edit]

Assume that is an odd prime, that   for pairwise relatively prime integers (i.e. in )   and that

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

(Wieferich 1909)[13][14] Under the above assumptions,  

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

(Mirimanoff 1911)[15] Under the above assumptions  

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)[16][17] Under the above assumptions, for every prime  

(Furtwängler 1912)[18] Under the above assumptions, for every prime  

(Vandiver)[19] Under the above assumptions, if in addition     then     and  

Powers mod most primes[edit]

Eisenstein's law can be used to prove the following theorem (Trost, Ankeny, Rogers).[20]   Suppose     and that     where     is an odd prime. If     is solvable for all but finitely many primes     then  

See also[edit]

Notes[edit]

  1. ^ Lemmermeyer, p. 392.
  2. ^ Ireland & Rosen, ch. 14.2
  3. ^ Lemmermeyer, ch. 11.2, uses the term semi-primary.
  4. ^ Ireland & Rosen, lemma in ch. 14.2 (first assertion only)
  5. ^ Lemmereyer, lemma 11.6
  6. ^ Ireland & Rosen, prop 13.2.7
  7. ^ Lemmermeyer, prop. 3.1
  8. ^ a b c Lemmermeyer, thm. 11.9
  9. ^ Ireland & Rosen, ch. 14 thm. 1
  10. ^ Ireland & Rosen, ch. 14.5
  11. ^ Lemmermeyer, ch. 11.2
  12. ^ Lemmermeyer, ch. 11 notes
  13. ^ Lemmermeyer, ex. 11.33
  14. ^ Ireland & Rosen, th. 14.5
  15. ^ Lemmermeyer, ex. 11.37
  16. ^ Lemmermeyer, ex. 11.32
  17. ^ Ireland & Rosen, th. 14.6
  18. ^ Lemmermeyer, ex. 11.36
  19. ^ Ireland & Rosen, notes to ch. 14
  20. ^ Ireland & Rosen, ch. 14.6, thm. 4. This is part of a more general theorem: Assume for all but finitely many primes Then i) if then but ii) if then or

References[edit]

  • Eisenstein, Gotthold (1850), "Beweis der allgemeinsten Reciprocitätsgesetze zwischen reellen und komplexen Zahlen", Verhandlungen der Königlich Preußische Akademie der Wissenschaften zu Berlin (in German): 189–198, Reprinted in Mathematische Werke, volume 2, pages 712–721 
  • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer Science+Business Media, ISBN 0-387-97329-X