Genera in Quadratic Fields and Their Character Sets

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.