Scholz's reciprocity law

In

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,
  Zbl 0949.11002
• Scholz, Arnold (1929), "Zwei Bemerkungen zum Klassenkörperturm.",
  JFM 55.0103.06
• Schönemann, Theodor (1839), "Ueber die Congruenz x² + y² ≡ 1 (mod p)",
  ERAM 019.0611cj