Abstract Algebra: An Introduction by Thomas W. Hungerford

By Thomas W. Hungerford

Summary ALGEBRA: AN creation is meant for a primary undergraduate path in smooth summary algebra. Its versatile layout makes it compatible for classes of varied lengths and diversified degrees of mathematical sophistication, starting from a conventional summary algebra path to at least one with a extra utilized taste. The booklet is prepared round subject matters: mathematics and congruence. every one subject is built first for the integers, then for polynomials, and at last for earrings and teams, so scholars can see the place many summary innovations come from, why they're vital, and the way they relate to 1 another.
New Features:
- A groups-first choice that allows those that are looking to disguise teams ahead of earrings to take action easily.
- Proofs for rookies within the early chapters, that are damaged into steps, each one of that's defined and proved in detail.
- within the center direction (chapters 1-8), there are 35% extra examples and thirteen% extra workouts.

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Mathématiques 1re S et E

Desk des matières :

Chapitre 1. L’outil vectoriel et analytique
    I. Introduction
    II. Le plan vectoriel (rappels)
    III. Les liaisons « plan ponctuel-plan vectoriel »
    IV. L’outil analytique
    V. Compléments
    Exercices

Chapitre 2. L’outil des transformations
    I. Introduction
    II. modifications usuelles
    III. motion sur les configurations élémentaires
    IV. changes associant une determine donnée à une determine donnée
    V. Composition de transformations
    VI. Compléments
    Exercices

Chapitre three. Les angles
    I. Introduction
    II. perspective d’un couple de vecteurs
    III. L’addition des angles
    IV. Propriétés géométriques
    V. Angles et cercles
    VI. Compléments
    Exercices

Chapitre four. Le produit scalaire
    I. Introduction
    II. Produit scalaire de deux vecteurs (rappel)
    III. Produit scalaire en géométrie analytique
    IV. Orthogonalité et cocyclicité
    V. Produit scalaire et lignes de niveau
    VI. Compléments
    Exercices

Chapitre five. Trigonométrie et family métriques dans le triangle
    I. Introduction
    II. Cosinus et sinus (rappels)
    III. Cosinus et produit scalaire ; sinus et déterminant
    IV. Trigonométrie
    V. family members métriques dans le triangle
    VI. Compléments
    Trigonométrie (formulaire récapitulatif)
    Exercices

Chapitre 6. Rotations et isométries fixant un aspect donné
    I. creation (quart de tour)
    II. Rotation de centre O et d’angle α
    III. Rotation : théorèmes de composition et propriétés géométriques
    IV. Isométries fixant un aspect donné
    V. Compléments
    Exercices

Chapitre 7. Le calcul vectoriel dans l’espace
    I. Introduction
    II. L’espace vectoriel E
    III. Droites et plans : repères et vecteurs directeurs
    IV. Éléments de géométrie analytique dans l’espace
    V. Compléments
    Exercices

Chapitre eight. Le produit scalaire dans l’espace
    I. Introduction
    II. Produit scalaire dans E
    III. purposes géométriques du produit scalaire
    IV. Produit scalaire et géométrie analytique
    V. Compléments
    Exercices

Chapitre nine. l. a. sphère
    I. Introduction
    II. l. a. sphère : définition et premières propriétés
    III. part d’une sphère
    IV. Détermination d’une sphère
    V. Surfaces de révolution
    VI. Compléments
    Exercices

Chapitre 10. Statistiques
    I. Introduction
    II. Les caractéristiques de position
    III. Les caractéristiques de dispersion
    IV. Compléments
    Exercices

Additional resources for Abstract Algebra: An Introduction

Example text

4 shows that solving equations in Z,, may be quite different from solving equations in 7L. A quadratic equation in 7L has at most two solutions, whereas the quadratic equation x1 ffi [5]0x = [OJ has four solutions in Z6• • Exercises A. I. Write out the addition and multiplication tables for (a) (b) � Z2 (c) 7L7 (d) Z-12 In Exercises 2--8, solve the equation. 2. _ .. eMmog-- .. _:ligl:U�:MpiNit. 3 3. x2 4. ls x4 =[lJ in Zs 5. x2 EB [3J 0 x®[2J = [OJ in Zt, 6. x2 EB [SJ 0 x = [OJin £9 7. x3 EB x2® x®[lJ =[OJ in Zs 8.

A). ] 31. If p is a positive prime, prove that Vfi is irrational. ] 32. (Euclid ) Prove that there are infinitely many primes. [Hint: Use proof by contradiction (Appendix A) . Assume there are only finitely many primes p1, p2, Pk• and reach a contradiction by showing that the number , Pk·l p1p2 Pk + 1 is not divisible by any of Pi. p2, 33. Let p > 1. If 2P - 1 is prime, prove that p is prime. ] • • • · , • • • · • Note: The converse is false by Exercise 2(b). C. 34. Prove or disprove: If n is an integer and n > 2, then there exists a prime p such that n

A2, , an are integers, not all zero, then their greatest common divisor (gcd) is the largest integer d such that d I a1for every i. Prove that there exist integers u1 such that d = a1u1 + a2u2 + + anu,.. [Hint: Adapt the proof of Theorem 1. ] • • • · • · 31. The least common multiple (lcm) of nonzero integers a1, � , ak is the smallest positive integer m such that a1lm for i = 1, 2, , k and is denoted [a1> � , ak1. • • • • ... •• • (a) • Find each of the following: [6, 10], [4, 5, 6, 10], (20, 42], and [2, 3, 14, 36, 42].

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