Lectures on Number Theory

Front Cover
American Mathematical Soc. - Mathematics - 275 pages
This volume is a translation of Dirichlet's Vorlesungen uber Zahlentheorie which includes nine supplements by Dedekind and an introduction by John Stillwell, who translated the volume.
Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions.
The book is suitable as a textbook, yet it also offers a fascinating historical perspective that links Gauss with modern number theory.
 

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Contents

Translators introduction
xi
On the divisibility of numbers
1
On the congruence of numbers
21
On quadratic residues
53
On quadratic forms
91
Determination of the class number of binary quadratic forms
149
Some theorems from Gausss theory of circle division
199
On the limiting value of an infinite series
211
A geometric theorem
215
Genera of quadratic forms
217
Power residues for composite moduli
229
Primes in arithmetic progressions
237
Some theorems from the theory of circle division
249
On the Pell equation
257
Convergence and continuity of some infinite series
261
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Page xiii - Legendre's statement of the theorem — which is to say that/? is a square mod q if and only if q is a square mod p...
Page 3 - The logarithms of negative factors in any group are found without regard to the negative signs, and the result is positive or negative according as the number of negative factors is even or odd.
Page xvi - It is greatly to be lamented that this virtue of the real numbers [that is, of the ordinary integers] to be decomposable into prime factors, always the same ones for a given number, does not also belong to the complex numbers...
Page xv - ... he was completing the section on composition of forms in the Disquisitiones Arithmeticae. but that he was never able to put it on a firm foundation; he says in particular, in a note to his article on the decomposition of polynomials into linear factors, that: 'If I wanted to proceed with the use of imaginaries in the way that earlier mathematicians have done, then one of my earlier researches which is very difficult could have been done in a very simple way.

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