Genera in Quadratic Fields and Their Character Sets

Abstract

For the further development of the theory of quadratic fields, particularly in order to classify the ideal classes of such a field, we make use of a new symbol. Let n and m be rational integers, with m not the square of an integer; let w be any rational prime number. Then the symbol $(\frac{{n,m}}{w})$ takes the value +1 whenever n is congruent modulo w to the norm of an integer of the quadratic field $k\left( {\sqrt m } \right)$ determined by $\sqrt m$ and in addition for each higher power of w there exists an integer in $k\left( {\sqrt m } \right)$ whose norm is congruent to n modulo the corresponding power of w; in every other situation we set $\left( {\frac{{n,m}}{w}} \right) = - 1$. The rational integers n for which $\left( {\frac{{n,m}}{w}} \right) = + 1$ are called norm residues of the field $k\left( {\sqrt m } \right)$ modulo w; the integers n for which $\left( {\frac{{n,m}}{w}} \right) = - 1$ are called norm non-residues of $k\left( {\sqrt m } \right)$ modulo w. If m is the square of an integer we shall always take $\left( {\frac{{n,m}}{w}} \right)$ to be +1. The following theorem gives us information concerning the properties of the symbol $\left( {\frac{{n,m}}{w}} \right)$ which are useful in calculations.