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Finite Abelian Groups and Their Characters,rithmetical functions called .. Although the study of Dirichlet characters can be undertaken without any knowledge of groups, the introduction of a minimal amount of group theory places the theory of Dirichlet characters in a more natural setting and simplifies some of the discussion.
,Dirichlet’s Theorem on Primes in Arithmetical Progressions,ssary condition for the existence of infinitely many primes in the arithmetic progression (1) is that (.) = 1. Dirichlet was the first to prove that this condition is also sufficient. That is, if (.) = 1 the arithmetic progression (1) contains infinitely many primes. This result, now known as ., will be proved in this chapter.
Partitions,ements of . is called a partition of . and we are interested in the arithmetical function .(.) which counts the number of partitions of . into summands taken from .. We illustrate with some famous examples.
Quadratic Residues and the Quadratic Reciprocity Law,ORbaOaacaaIWaaaaa!42A7!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$n
ot equiv 0$$ (mod .). Since the modulus is prime we know that (1) has at most two solutions. Moreover, if . is a solution so is − ., hence the number of solutions is either 0 or 2.
The Fundamental Theorem of Arithmetic, principal results are Theorem 1.2, which establishes the existence of the greatest common divisor of any two integers, and Theorem 1.10 (the fundamental theorem of arithmetic), which shows that every integer greater than 1 can be represented as a product of prime factors in only one way (apart from
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,Dirichlet’s Theorem on Primes in Arithmetical Progressions,essions have this property. An arithmetic progression with first term . and common difference . consists of all numbers of the form .If . and . have a common factor .,each term of the progression is divisible by . and there can be no more than one prime in the progression if ..In other words, a nece