Download e-book for iPad: Abstract Algebra: A Study Guide for Beginners by Paul T. Bateman

By Paul T. Bateman

ISBN-10: 9812389385

ISBN-13: 9789812389381

I first taught an summary algebra direction in 1968. utilizing Hcrstein's themes in Algebra. it really is challenging to enhance on his publication; the topic can have turn into broader, with purposes to computing and different parts, yet subject matters includes the middle of any direction. regrettably, the topic hasn't turn into any more uncomplicated, so scholars assembly summary algebra nonetheless fight to profit the recent recommendations, in particular in view that they're most likely nonetheless studying tips to write their very own proofs.This "study advisor" is meant to aid scholars who're commencing to find out about summary algebra. rather than simply increasing the cloth that's already written down in our textbook, i made a decision to attempt to coach via instance, by means of writing out recommendations to difficulties. i have attempted to decide on difficulties that may be instructive, and in a number of circumstances i have integrated reviews to assist the reader see what's particularly happening. after all, this examine advisor is not an alternative to a great instructor, or for the opportunity to interact with different scholars on a few difficult problems.Finally. i want to gratefully recognize the help of Northern Illinois college whereas scripting this learn consultant. As a part of the popularity as a "Presidential instructing Professor," i used to be given depart in Spring 2000 to paintings on tasks regarding instructing.

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Desk des matières du livre :

Chapitre I. — Opérations sur les nombres entiers
    1. Les nombres entiers
    2. Addition. Somme
    3. Multiplication. Produit
    4. Suites d’additions et de multiplications. — utilization des parenthèses
    5. Produits de sommes
    6. Pratique de l’addition
    7. Pratique de los angeles multiplication
    8. Soustraction. Différence
    9. Polynômes arithmétiques
    10. Produits de différences
    11. Pratique de l. a. soustraction
    12. Multiples et diviseurs d’un nombre. — Quotient exact
    13. Multiples et diviseurs d’un nombre (suite)
    14. Quotient de deux nombres à une unité près

Chapitre II. — Divisibilité
    15. Divisibilité par 2 et 5 ; par four et 25
    16. Divisibilité par nine et par 3
    17. Multiples et diviseurs communs à deux ou plusieurs nombres
    18. Nombres premiers
    19. Décomposition d’un nombre en un produit de facteurs premiers
    20. Recherche des diviseurs d’un nombre
    21. Plus grand commun diviseur
    22. Plus petit commun multiple
    23. Nombres premiers entre eux
    Problèmes sur le chapitre II

Chapitre III. — Fractions et nombres décimaux
    24. inspiration de fraction
    25. Fractions égales
    26. Simplification des fractions
    27. Réduction des fractions au même dénominateur
    28. Multiplication des fractions
    29. department des fractions
    30. Addition des fractions
    31. Comparaison des fractions. — Soustraction des fractions
    32. Opérations sur les sommes, différences et produits de nombres entiers ou fractionnaires
    33. Fractions décimales. — Nombres décimaux
    34. Opérations sur les nombres décimaux
    35. Quotient de deux nombres à une unité décimale près
    36. Fractions ordinaires et nombres décimaux
    Problèmes sur le chapitre III

Chapitre IV. — Nombres complexes
    37. Nombres complexes

Chapitre V. — Arithmétique littérale. — Résolution algébrique des problèmes
    38. Expressions littérales
    39. Égalités et équations
    40. Résolution algébrique des problèmes

Chapitre VI. — Racine carrée
    41. Racine carrée
    42. Recherche de los angeles racine carrée d’un nombre
    43. Extraction de los angeles racine carrée d’un nombre
    Problèmes sur le chapitre VI

Chapitre VII. — Rapports et proportions. — Applications
    44. Rapport de deux nombres
    45. Proportions
    46. Suite de rapports égaux. — Nombres proportionnels
    47. Grandeurs proportionnelles
    48. Pourcentages et bénéfices
    49. Intérêts simples
    50. Escompte commercial
    Problèmes sur le chapitre VII

Additional resources for Abstract Algebra: A Study Guide for Beginners

Example text

Given (a1 , b1 ), (a2 , b2 ), and (a3 , b3 ) with (a1 , b1 ) ∼ (a2 , b2 ) and (a2 , b2 ) ∼ (a3 , b3 ), we have the equations a1 b2 = a2 b1 and a2 b3 = a3 b2 . If we multiply the first equation by b3 and the second equation by b1 , we get a1 b2 b3 = a2 b1 b3 = a3 b1 b2 . Since b2 = 0 we can cancel to obtain a1 b3 = a3 b1 , showing that (a1 , b1 ) ∼ (a3 , b3 ). 15. On the set C of complex numbers, define z1 ∼ z2 if ||z1 || = ||z2 ||. Show that ∼ is an equivalence relation. Solution: The reflexive, symmetric, and transitive laws can be easily verified since ∼ is defined in terms of an equality, and equality is itself an equivalence relation.

Solve the system of congruences 5x ≡ 14 (mod 17) 3x ≡ 2 (mod 13) . Solution: By trial and error, 7 · 5 ≡ 1 (mod 17) and 9 · 3 ≡ 1 (mod 13), so 5x ≡ 14 (mod 17); 35x ≡ 98 (mod 17); x ≡ 13 (mod 17) and 3x ≡ 2 (mod 13); 27x ≡ 18 (mod 13); x ≡ 5 (mod 13). Having reduced the system to the standard form, we can solve it in the usual way. We have x = 13 + 17q for some q ∈ Z, and then 13 + 17q ≡ 5 (mod 13). This reduces to 4q ≡ 5 (mod 13), so 40q ≡ 50 (mod 13), or q ≡ 11 (mod 13). This leads to the answer, x ≡ 13 + 17 · 11 ≡ 200 (mod 221).

A) Compute the last digit in the decimal expansion of 4100 . Solution: The last digit is the remainder when divided by 10. Thus we must compute the congruence class of 4100 (mod 10). We have 42 ≡ 6 (mod 10), and then 62 ≡ 6 (mod 10). Thus 4100 = (42 )50 ≡ 650 ≡ 6 (mod 10). (b) Is 4100 divisible by 3? Solution: No, since 4100 ≡ 1100 ≡ 1 (mod 3). Or you can write 2200 as the prime factorization, and then (3, 2200 ) = 1. 34. Find all integers n for which 13 | 4(n2 + 1). Solution: This is equivalent solving the congruence 4(n2 + 1) ≡ 0 (mod 13).

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Abstract Algebra: A Study Guide for Beginners by Paul T. Bateman


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