1911 Encyclopædia Britannica/Number/Characters of a form or class
|←Number/Automorphs||1911 Encyclopædia Britannica, Volume 19
- Number Characters of a form or class
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36. Characters of a form or class. Genera.—Let be any primitive form; we have seen above (§ 32) that if are any integers
where . Now the expressions in brackets on the left hand may denote any two numbers representable by the form ; the formula shows that is a residue of , and hence is a residue of every odd prime factor of , and if is any such factor the symbols and will have the same value. Putting , this common value is denoted by and called a quadratic character (or simply character) of with respect to . Since is representable by () the value is the same as . For example, if , the scheme of characters for the six reduced primitive forms, and therefore for the classes they represent, is
In certain cases there are supplementary characters of the type and , and the characters are discriminated according as an odd or even power of is contained in ; but in every case there are certain combinations of characters (in number one-half of all possible combinations) which form the total characters of actually existing classes. Classes which have the same total character are said to belong to the same genus. Each genus of the same order contains the same number of classes.
For any determinant we have a principal primitive class for which all the characters are ; this is represented by the principal form or according as is of the form or . The corresponding genus is called the principal genus. Thus, when , it appears from the table above that in the primitive order there are two genera, each containing three classes; and the non-existent total characters are and .