Genera in Quadratic Fields and Their Character Sets

  • David Hilbert


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.


Prime Number Ideal Class Fundamental Unit Quadratic Field Quadratic Residue 
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© Springer-Verlag Berlin Heidelberg 1998

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  • David Hilbert

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