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A Concrete Introduction to Higher Algebra de Lindsay N. Childs

A Concrete Introduction to Higher Algebra: Undergraduate Texts in Mathematics

Autor Lindsay N. Childs
en Limba Engleză Paperback – 14 ian 2000

Nivel de studiu: licență. Notăm cu interes apariția celei de-a treia ediții a lucrării A Concrete Introduction to Higher Algebra, semnată de Lindsay N Childs, un volum ce reușește să transforme abstractizarea algebrei superioare într-un parcurs intuitiv pentru studenții aflați la primul contact cu domeniul. Merită menționat că autorul alege o abordare inductivă, pornind de la structuri familiare — numerele întregi și polinoamele — pentru a introduce riguros conceptele de inel și corp. Putem afirma că această metodă facilitează tranziția de la calculul matematic de bază către gândirea structurală specifică algebrei moderne. Organizarea volumului reflectă o progresie logică, de la algoritmul lui Euclid și factorizarea unică, până la teme complexe precum reciprocitatea pătratică și clasificarea corpurilor finite. Un element distinctiv îl reprezintă integrarea masivă a aplicațiilor practice; teoria nu rămâne izolată, ci este utilizată pentru a explica mecanismele din spatele criptografiei și teoriei codurilor (BCH). Comparabil cu A First Course in Abstract Algebra de Joseph J. Rotman în ceea ce privește rigurozitatea, volumul lui Lindsay N Childs se diferențiază prin accentul pus pe latura computațională și prin includerea unor algoritmi moderni, precum testul de primalitate probabilist al lui Rabin. Cu peste 900 de exerciții distribuite strategic, textul încurajează participarea activă a cititorului. Cele peste 600 de pagini publicate de Springer oferă un suport didactic solid, unde demonstrațiile teoretice sunt echilibrate constant de contextul lor istoric și de utilitatea lor în securitatea datelor.

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Specificații

ISBN-13: 9780387989990
ISBN-10: 0387989994
Pagini: 537
Ilustrații: XV, 522 p.
Dimensiuni: 155 x 235 x 30 mm
Greutate: 0.89 kg
Ediția:2nd ed. 1995. 2nd printing 2000
Editura: Springer
Colecția Springer
Seria Undergraduate Texts in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Lower undergraduate

De ce să citești această carte

Recomandăm această carte studenților la matematică și informatică de la nivel licență care doresc să înțeleagă fundamentul algebric al criptografiei moderne. Cititorul câștigă o bază solidă în teoria numerelor și algebră abstractă prin exemple concrete, nu doar prin teoreme izolate. Este o resursă esențială datorită celor 900 de exerciții și a explicațiilor clare asupra modului în care inelele și corpurile finite sunt aplicate în securitatea digitală.

Descriere scurtă

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The first edition of this book emerged from a set of notes written in the 1970sfor a sophomore-junior level course at the University at Albany entitled "Classical Algebra." The objective of the course, and the book, is to give students enough experience in the algebraic theory of the integers and polynomials to appre­ ciate the basic concepts of abstract algebra. The main theoretical thread is to develop algebraic properties of the ring of integers: unique factorization into primes, congruences and congruence classes, Fermat's theorem, the Chinese remainder theorem; and then again for the ring of polynomials. Doing so leads to the study of simple field extensions, and, in particular, to an exposition of finite fields. Elementary properties of rings, fields, groups, and homomorphisms of these objects are introduced and used as needed in the development. Concurrently with the theoretical development, the book presents a broad variety of applications, to cryptography, error-correcting codes, Latin squares, tournaments, techniques of integration, and especially to elemen­ tary and computational number theory. A student who asks, "Why am I learning this?," willfind answers usually within a chapter or two. For a first course in algebra, the book offers a couple of advantages. • By building the algebra out of numbers and polynomials, the book takes maximal advantage of the student's prior experience in algebra and arithmetic. New concepts arise in a familiar context.

Cuprins

1 Numbers.- 2 Induction.- A. Induction.- B. Another Form of Induction.- C. Well-Ordering.- D. Division Theorem.- E. Bases.- F. Operations in Base a.- 3 Euclid’s Algorithm.- A. Greatest Common Divisors.- B. Euclid’s Algorithm.- C. Bezout’s Identity.- D. The Efficiency of Euclid’s Algorithm.- E. Euclid’s Algorithm and Incommensurability.- 4 Unique Factorization.- A. The Fundamental Theorem of Arithmetic.- B. Exponential Notation.- C. Primes.- D. Primes in an Interval.- 5 Congruences.- A. Congruence Modulo m.- B. Basic Properties.- C. Divisibility Tricks.- D. More Properties of Congruence.- E. Linear Congruences and Bezout’s Identity.- 6 Congruence Classes.- A. Congruence Classes (mod m): Examples.- B. Congruence Classes and ?/m?.- C. Arithmetic Modulo m.- D. Complete Sets of Representatives.- E. Units.- 7 Applications of Congruences.- A. Round Robin Tournaments.- B. Pseudorandom Numbers.- C. Factoring Large Numbers by Trial Division.- D. Sieves.- E. Factoring by the Pollard Rho Method.- F. Knapsack Cryptosystems.- 8 Rings and Fields.- A. Axioms.- B. ?/m?.- C. Homomorphisms.- 9 Fermat’s and Euler’s Theorems.- A. Orders of Elements.- B. Fermat’s Theorem.- C. Euler’s Theorem.- D. Finding High Powers Modulo m.- E. Groups of Units and Euler’s Theorem.- F. The Exponent of an Abelian Group.- 10 Applications of Fermat’s and Euler’s Theorems.- A. Fractions in Base a.- B. RSA Codes.- C. 2-Pseudoprimes.- D. Trial a-Pseudoprime Testing.- E. The Pollard p — 1 Algorithm.- 11 On Groups.- A. Subgroups.- B. Lagrange’s Theorem.- C. A Probabilistic Primality Test.- D. Homomorphisms.- E. Some Nonabelian Groups.- 12 The Chinese Remainder Theorem.- A. The Theorem.- B. Products of Rings and Euler’s ?-Function.- C. Square Roots of 1 Modulo m.- 13 Matricesand Codes.- A. Matrix Multiplication.- B. Linear Equations.- C. Determinants and Inverses.- D. Mn(R).- E. Error-Correcting Codes, I.- F. Hill Codes.- 14 Polynomials.- 15 Unique Factorization.- A. Division Theorem.- B. Primitive Roots.- C. Greatest Common Divisors.- D. Factorization into Irreducible Polynomials.- 16 The Fundamental Theorem of Algebra.- A. Rational Functions.- B. Partial Fractions.- C Irreducible Polynomials over ?.- D. The Complex Numbers.- E. Root Formulas.- F. The Fundamental Theorem.- G. Integrating.- 17 Derivatives.- A. The Derivative of a Polynomial.- B. Sturm’s Algorithm.- 18 Factoring in ?[x], I.- A. Gauss’s Lemma.- B. Finding Roots.- C. Testing for Irreducibility.- 19 The Binomial Theorem in Characteristic p.- A. The Binomial Theorem.- B. Fermat’s Theorem Revisited.- C. Multiple Roots.- 20 Congruences and the Chinese Remainder Theorem.- A. Congruences Modulo a Polynomial.- B. The Chinese Remainder Theorem.- 21 Applications of the Chinese Remainder Theorem.- A. The Method of Lagrange Interpolation.- B. Fast Polynomial Multiplication.- 22 Factoring in Fp[x] and in ?[x].- A. Berlekamp’s Algorithm.- B. Factoring in ?[x] by Factoring mod M.- C. Bounding the Coefficients of Factors of a Polynomial.- D. Factoring Modulo High Powers of Primes.- 23 Primitive Roots.- A. Primitive Roots Modulo m.- B. Polynomials Which Factor Modulo Every Prime.- 24 Cyclic Groups and Primitive Roots.- A. Cyclic Groups.- B. Primitive Roots Modulo pe.- 25 Pseudoprimes.- A. Lots of Carmichael Numbers.- B. Strong a-Pseudoprimes.- C. Rabin’s Theorem.- 26 Roots of Unity in ?/m?.- A. For Which a Is m an a-Pseudoprime?.- B. Square Roots of ?1 in ?/p?.- C. Roots of ?1 in ?/m?.- D. False Witnesses.- E. Proof of Rabin’s Theorem.- F. RSA Codes andCarmichael Numbers.- 27 Quadratic Residues.- A. Reduction to the Odd Prime Case.- B. The Legendre Symbol.- C. Proof of Quadratic Reciprocity.- D. Applications of Quadratic Reciprocity.- 28 Congruence Classes Modulo a Polynomial.- A. The Ring F[x]/m(x).- B. Representing Congruence Classes mod m(x).- C. Orders of Elements.- D. Inventing Roots of Polynomials.- E. Finding Polynomials with Given Roots.- 29 Some Applications of Finite Fields.- A. Latin Squares.- B. Error Correcting Codes.- C. Reed-Solomon Codes.- 30 Classifying Finite Fields.- A. More Homomorphisms.- B. On Berlekamp’s Algorithm.- C. Finite Fields Are Simple.- D. Factoring xpn — x in Fp[x].- E. Counting Irreducible Polynomials.- F. Finite Fields.- G. Most Polynomials in Z[x] Are Irreducible.- Hints to Selected Exercises.- References.

Recenzii

From the reviews:
"The user-friendly exposition is appropriate for the intended audience. Exercises often appear in the text at the point they are relevant, as well as at the end of the section or chapter. Hints for selected exercises are given at the end of the book. There is sufficient material for a two-semester course and various suggestions for one-semester courses are provided. Although the overall organization remains the same in the second edition¿Changes include the following: greater emphasis on finite groups, more explicit use of homomorphisms, increased use of the Chinese remainder theorem, coverage of cubic and quartic polynomial equations, and applications which use the discrete Fourier transform." MATHEMATICAL REVIEWS

Textul de pe ultima copertă

This book is an informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials. A strong emphasis on congruence classes leads in a natural way to finite groups and finite fields. The new examples and theory are built in a well-motivated fashion and made relevant by many applications - to cryptography, error correction, integration, and especially to elementary and computational number theory. The later chapters include expositions of Rabin's probabilistic primality test, quadratic reciprocity, the classification of finite fields, and factoring polynomials over the integers. Over 1000 exercises, ranging from routine examples to extensions of theory, are found throughout the book; hints and answers for many of them are included in an appendix.
The new edition includes topics such as Luhn's formula, Karatsuba multiplication, quotient groups and homomorphisms, Blum-Blum-Shub pseudorandom numbers, root bounds for polynomials, Montgomery multiplication, and more.
"At every stage, a wide variety of applications is presented...The user-friendly exposition is appropriate for the intended audience"
- T.W. Hungerford, Mathematical Reviews
"The style is leisurely and informal, a guided tour through the foothills, the guide unable to resist numerous side paths and return visits to favorite spots..."
- Michael Rosen, American Mathematical Monthly

Caracteristici

Informal and readable introduction to higher algebra New sections on Luhn's formula, Cosets and equations, and detaching coefficients Successful undergraduate text for more than 20 years