The Theory of Algebraic Number Fields pp 121-132 | Cite as

# 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.

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