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Genera in Quadratic Fields and Their Character Sets

  • David Hilbert
Chapter

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.

Keywords

Prime Number Ideal Class Fundamental Unit Quadratic Field Quadratic Residue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • David Hilbert

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