Genera in Quadratic Fields and Their Character Sets

David Hilbert

Chapter 17 in The Theory of Algebraic Number Fields, 1998, pp 121-132 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1007/978-3-662-03545-0_17

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