A classical introduction to modern number theory
Bridging the gap between elementary number theory and the systematic study of advanced topics, A Classical Introduction to Modern Number Theory is a well-developed and accessible text that requires only a familiarity with basic abstract algebra. Historical development is stressed throughout, along with wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves.
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a e F algebraic integers algebraic number field assertion assume Bernoulli numbers biquadratic Chapter character of order class number coefficients complex numbers congruence conjecture consider Corollary cubic reciprocity curve cyclic defined definition degree denote Dirichlet character divides divisors Eisenstein equation Exercise Fermat's finite field formula Galois Gauss sums hypersurface implies infinitely many primes irreducible polynomials Jacobi sums law of quadratic Legendre symbol Lemma Let F minimal polynomial monic irreducible nonresidue nontrivial nonzero number of elements number of points number of solutions number theory odd prime ordp points at infinity positive integer primary prime ideal prime number Proposition prove q elements Q(Cm quadratic reciprocity quadratic residue quadratic residue mod rational prime reciprocity law relatively prime residue class result follows Riemann hypothesis ring of integers root of unity Section solvable Suppose Theorem Z[co Z/mZ Z/pZ zero zeta function