Lectures on Number Theory
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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A geometric theorem
Genera of quadratic forms
Power residues for composite moduli
Primes in arithmetic progressions
Some theorems from the theory of circle division
On the Pell equation
Convergence and continuity of some infinite series
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absolute value according assume belong called class number clear coefficients complete condition congruence Consequently consider continued fraction Conversely corresponding D.A. art decomposition denote derived Dirichlet 2nd edition distinct dividing divisible easily equal equation equivalent exactly example exponent expression fact factors finite follows footnote from Dirichlet formula four function given gives greatest common divisor hence immediately implies important incongruent kind latter least less limiting value modulus multiple namely negative nonresidue nonzero obtain obviously odd prime pairs particular period positive integer possible prime numbers problem proof proved quadratic residue reduced forms relatively prime remains representations represented respectively result roots runs satisfy sending smallest solution square substitution symbol tends theorem theory third transformation zero
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 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.