Scholz's reciprocity law

In mathematics, Scholz's reciprocity law is a reciprocity law for quadratic residue symbols of real quadratic number fields discovered by Theodor Schönemann (1839) and rediscovered by Arnold Scholz (1929).

Statement

Suppose that p and q are rational primes congruent to 1 mod 4 such that the Legendre symbol (p/q) is 1. Then the ideal (p) factorizes in the ring of integers of Q(√q) as (p)=𝖕𝖕' and similarly (q)=𝖖𝖖' in the ring of integers of Q(√p). Write ε_p and ε_q for the fundamental units in these quadratic fields. Then Scholz's reciprocity law says that

[ε_p/𝖖] = [ε_q/𝖕]

where [] is the quadratic residue symbol in a quadratic number field.

References

• Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Springer-Verlag, Berlin, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
• Scholz, Arnold (1929), "Zwei Bemerkungen zum Klassenkörperturm.", Journal für die reine und angewandte Mathematik (in German), 161: 201–207, doi:10.1515/crll.1929.161.201, ISSN 0075-4102, JFM 55.0103.06
• Schönemann, Theodor (1839), "Ueber die Congruenz x² + y² ≡ 1 (mod p)", Journal für die reine und angewandte Mathematik, 19: 93–112, doi:10.1515/crll.1839.19.93, ISSN 0075-4102, ERAM 019.0611cj