Elementary Number Theory: An Algebraic Approach
This 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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algebraic integers algorithm analogue Appendix argument arithmetic associates Chapter Chinese remainder theorem coeﬂicients computing congruence modulo conjugate Corollary 26.2 coset cyclic groups decimal expansion deﬁned Deﬁnition Diophantine equation x2 enjoys unique factorization equivalent solutions Euclidean domain Euclidean norm Euler Euler’s theorem example exponent fact Fermat conjecture ﬁeld ﬁnd ﬁnding ﬁnite ﬁrst function fundamental unit Gaussian integers greatest common divisor hence homomorphism ideal implies improper unit inertial inﬁnitely many primes integral domain isomorphism lattice points Lemma Let g mathematics N(ot nonzero number theory odd prime polynomial positive integers prime in A(m primitive Pythagorean triple primitive root Problem product of primes proof of Theorem Prove Pythagorean triple Quadratic Reciprocity ramiﬁes rational integers rational numbers rational primes congruent relatively prime representable integers residue ring satisﬁes Section sequence solutions to Eq Solve the Diophantine solves Eq square free subgroup subring subset Suppose true unique factorization domain write