## A concrete approach to abstract algebra |

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Page 80

Thus e has been expressed in the form ax + fry where x and ^ are

does not disturb us that x happens to be negative. The letter h is frequently used

for the H.C.F. Using this notation, we have the important theorem 3. theorem 3 . If

h is the H.C.F. of the

example, the numbers 31 and 40 have H.C.F. 1. Thus it must be possible to find

whole numbers x, y for which 31* + 40y = 1. To find these by trial and error would

not ...

Thus e has been expressed in the form ax + fry where x and ^ are

**integers**. Itdoes not disturb us that x happens to be negative. The letter h is frequently used

for the H.C.F. Using this notation, we have the important theorem 3. theorem 3 . If

h is the H.C.F. of the

**integers**a, b, thtt* exist**integers**x, y such that h = ax + by. Forexample, the numbers 31 and 40 have H.C.F. 1. Thus it must be possible to find

whole numbers x, y for which 31* + 40y = 1. To find these by trial and error would

not ...

Page 88

Note that we call an element prime when it has no factors within the structure. For

example, 3 = (V7 - 2)(V7 + 2) and *2 - 2 = (x - V2)(x + V2). But V7 — 2 and V7 + 2

are not

coefficients. 3 and x2 — 2 are prime elements of the

over rationals respectively. Highest Common Factor d is a common factor of a

and b if a = /></, 6 = ?rf, with /> and 9 elements of the structure. We now have to

explain what ...

Note that we call an element prime when it has no factors within the structure. For

example, 3 = (V7 - 2)(V7 + 2) and *2 - 2 = (x - V2)(x + V2). But V7 — 2 and V7 + 2

are not

**integers**, and x — Vl and x + V2 are not polynomials with rationalcoefficients. 3 and x2 — 2 are prime elements of the

**integers**and of polynomialsover rationals respectively. Highest Common Factor d is a common factor of a

and b if a = /></, 6 = ?rf, with /> and 9 elements of the structure. We now have to

explain what ...

Page 155

K: the

of a space of 2 dimensions with {a, b) as*the label for a + \b. But then (3, 0), (2, 2),

(1, 4),*(0, 6), and many others would all label the same thing. In fact, of course,

whatever

_i nil 14 1 ... These are all of the form §m where m is an

of 1 dimension over the

K: the

**integers**. Things: a + §£, (a, b**integers**). This has perhaps the appearanceof a space of 2 dimensions with {a, b) as*the label for a + \b. But then (3, 0), (2, 2),

(1, 4),*(0, 6), and many others would all label the same thing. In fact, of course,

whatever

**integers**you choose for a, b, the value of a + $b will lie in the set ... _ii _i_i nil 14 1 ... These are all of the form §m where m is an

**integer**. We have a spaceof 1 dimension over the

**integers**. Basis, the single Thing, \. (m) is the label for \m.### What people are saying - Write a review

#### LibraryThing Review

User Review - bookaholixanon - LibraryThingSee my review of Mathematician's Delight. Basically, anything written by W. W. Sawyer is pure gold, and worth reading. This book shows that Sawyer is just as adept with advanced as with elementary material. Read full review

### Contents

Introduction | 1 |

The Viewpoint of Abstract Algebra | 5 |

Arithmetics and Polynomials | 26 |

Copyright | |

11 other sections not shown

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### Common terms and phrases

Abstract Algebra Accordingly answer apply arith arithmetic modulo axiom 70 belongs blue system calculating machine chapter class containing complex numbers consider constant polynomial corresponds course cubic cubic equation defined definition dimensions over F displacement divided division earlier elementary algebra elements of F equa example fact field axioms field F form a basis form a field gives Hence inches East integers irreducible polynomial isomorphic label leave remainder linear expressions linearly dependent mathematics means ments metic mixture modular arithmetics modulo x2 natural numbers negative numbers nomial obtain odd number operations ordinary arithmetic plane poly possible procedure proof properties prove question rational coefficients rational numbers real numbers result satisfies an equation screws sequence simpler solution square root standard form statement structure subtract Suppose symbol theorem things three vectors tion vector space whole number write zero vector