# Artin's law of reciprocity

From a historical perspective, Artin's law of reciprocity (after Emil Artin ) comprised all previously known reciprocity laws, such as the quadratic reciprocity law . It says that a quotient of a generalized ideal class group of an Abelian field extension is isomorphic to the Galois group of this extension.

Artin's law of reciprocity is an essential step on the way to the solution of Hilbert's ninth problem and, because of its importance, is also called the main theorem of class field theory.

It can be formulated more precisely as follows:

${\ displaystyle I ({\ mathfrak {m}}) / P _ {\ mathfrak {m}} {\ mathfrak {N}} ({\ mathfrak {m}}) \ simeq G (K / k)}$

Here, the quantity of the module of prime ideals of , the group of standards of broken ideals in relatively prime to and the subset of (Group of broken main Ideal ), which from the broken main ideals consists with , wherein a subset of the unit group is. The explanatory module must be divisible by all branched prime ideals. ${\ displaystyle I ({\ mathfrak {m}})}$ ${\ displaystyle {\ mathfrak {m}}}$ ${\ displaystyle k}$${\ displaystyle {\ mathfrak {N}} ({\ mathfrak {m}})}$${\ displaystyle K}$${\ displaystyle {\ mathfrak {m}}}$${\ displaystyle P _ {\ mathfrak {m}}}$${\ displaystyle P}$${\ displaystyle (\ alpha)}$${\ displaystyle \ alpha \ in k _ {\ mathfrak {m}}}$${\ displaystyle k _ {\ mathfrak {m}}}$ ${\ displaystyle k ^ {\ times}}$${\ displaystyle {\ mathfrak {m}}}$

In terms of adel theory , it can be formulated as follows:

${\ displaystyle H ^ {3} (G (K / k), \ mathbb {A} _ {K}) \ simeq G (K / k)}$