Algebraic Number Theory
From its history as an elegant but abstract area of mathematics, algebraic number theory now takes its place as a useful and accessible study with important real-world practicality. Unique among algebraic number theory texts, this important work offers a wealth of applications to cryptography, including factoring, primality-testing, and public-key cryptosystems.
A follow-up to Dr. Mollin's popular Fundamental Number Theory with Applications, Algebraic Number Theory provides a global approach to the subject that selectively avoids local theory. Instead, it carefully leads the student through each topic from the level of the algebraic integer, to the arithmetic of number fields, to ideal theory, and closes with reciprocity laws. In each chapter the author includes a section on a cryptographic application of the ideas presented, effectively demonstrating the pragmatic side of theory.
In this way Algebraic Number Theory provides a comprehensible yet thorough treatment of the material. Written for upper-level undergraduate and graduate courses in algebraic number theory, this one-of-a-kind text brings the subject matter to life with historical background and real-world practicality. It easily serves as the basis for a range of courses, from bare-bones algebraic number theory, to a course rich with cryptography applications, to a course using the basic theory to prove Fermat's Last Theorem for regular primes. Its offering of over 430 exercises with odd-numbered solutions provided in the back of the book and, even-numbered solutions available a separate manual makes this the ideal text for both students and instructors.
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Arithmetic of Number Fields
Ideal Decomposition in Extension Fields
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abelian group algebraic integer algebraic number field algebraic number theory algorithm assume basis for F Biquadratic Reciprocity called Chapter Claim commutative ring completely congruence conjugates contradiction Corollary cubic cyclotomic Dedekind domain defined denoted Df-ideal discriminant divisor elliptic curve embeddings of F Equation establish Euclidean domain Example Exercise exists extension of number extension of Q field F field sieve finite field Footnote Galois extension given in Definition Hence homomorphism induction integral basis integral domain irreducible isomorphism Kummer Lemma Let F Let K/F mathematics matrix minimal polynomial multiplicative namely natural numbers NF(a nontrivial norm notion number field sieve Of-ideal Op-ideal primary prime D^-ideal prime ideals prime Ok-ideal primitive Prove Q(CP R-ideal ramified rational integer rational prime reader Reciprocity Law regular primes relatively prime result ring with identity root of unity Section solution subfield subgroup Suppose unique factorization unramified Z-module
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Codes: The Guide to Secrecy From Ancient to Modern Times
Richard A. Mollin
Limited preview - 2005