In mathematics, Scholz's reciprocity law is a

reciprocity law In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...

for quadratic residue symbols of real

quadratic number field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...

s discovered by and rediscovered by .

Statement

Suppose that ''p'' and ''q'' are rational

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

s congruent to 1 mod 4 such that the

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue ...

(''p''/''q'') is 1. Then the ideal (''p'') factorizes in the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...

of Q() as (''p'')=𝖕𝖕' and similarly (''q'')=𝖖𝖖' in the ring of integers of Q(). Write ε_''p'' and ε_''q'' for the fundamental units in these quadratic fields. Then Scholz's reciprocity law says that : �_''p''/𝖖= �_''q''/𝖕 where [] is the quadratic residue symbol in a quadratic number field.

References

* * *{{Citation , last1=Schönemann , first1=Theodor , title=Ueber die Congruenz x² + y² ≡ 1 (mod p) , url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002141868 , year=1839 , journal=

Journal für die reine und angewandte Mathematik ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English language, English: ''Journal for Pure and Applied Mathematics''). History The journal wa ...

, issn=0075-4102 , volume=19 , pages=93–112 , doi=10.1515/crll.1839.19.93 , id={{ERAM, 019.0611cj Theorems in algebraic number theory