Scholz's reciprocity law
Encyclopedia
In mathematics, Scholz's reciprocity law is a reciprocity law
for quadratic residue symbols of real quadratic number fields discovered by and rediscovered by .
Write εp and εq for the fundamental units in these quadratic fields. Then Scholz's reciprocity law says that
where [] is the quadratic residue symbol in a quadratic number field.
Reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity.There are several different ways to express reciprocity laws...
for quadratic residue symbols of real quadratic number fields discovered by and rediscovered by .
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' inthe ring of integers of Q(√p).Write εp and εq for the fundamental units in these quadratic fields. Then Scholz's reciprocity law says that
- [εp/𝖖] = [εq/𝖕]
where [] is the quadratic residue symbol in a quadratic number field.