Artin reciprocity law

Artin reciprocity law is a fundamental principle in number theory that connects prime numbers in different number systems. It predicts how prime factors in one type of number field relate to those in another, establishing a deep link between field extensions and their arithmetic properties. Essentially, it generalizes quadratic reciprocity (which explains how primes behave with respect to squares) to more complex settings, providing a unified framework for understanding the solvability of equations and the behavior of primes across various algebraic contexts. This law is central to class field theory, helping mathematicians classify and analyze abelian extensions of number fields.