Elementary Number Theory: An Algebraic ApproachThis text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and the Fermat conjecture for exponents three and four. The text contains abundant numerical examples and a particularly helpful collection of exercises, many of which are small research problems requiring substantial study or outside reading. Some problems call for new proofs for theorems already covered or for inductive explorations and proofs of theorems found in later chapters. Ethan D. Bolker teaches mathematics and computer science at the University of Massachusetts, Boston. |
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a₁ algebraic integers algorithm Appendix arithmetic associates b₁ Chinese remainder theorem coefficients computing congruence modulo Corollary 26.2 coset cyclic groups decimal expansion Definition Diophantine equation x² enjoys unique factorization Euclidean domain Euclidean norm Euler example exponent Fermat conjecture finite function Gaussian integers greatest common divisor hence homomorphism implies improper unit inertial infinitely many primes integral domain isomorphism lattice points Lemma Let g MATHEMATICS mb² modulo my² N₁ nonzero Number Fields number theory odd prime Pell's equation polynomial positive integers prime in A(m primitive root Problem product of primes proof of Theorem Prove Pythagorean triple Quadratic Reciprocity r₁ ramifies rational integers rational numbers rational primes congruent relatively prime residue ring Section solutions to Eq solves Eq square free subgroup subring Suppose unique factorization domain write