College or university Algebra

 College Algebra Essay

Introduction to Modern Algebra

David Joyce

Clark University

Version zero. 0. six, 3 April 2008

you Copyright

(C) 2008.

you

ii

I actually dedicate this book to my friend and colleague Arthur Noir. Arthur motivated me to publish this book. I apologize that this individual did not live to see that finished.

Contents

1 Advantages

1 . one particular Structures in Modern Algebra......................... 1 . 1 . 1 Businesses on sets........................... 1 . 1 . 2 Fields.................................. 1 . 1 ) 3 Rings.................................. 1 . 1 ) 4 Groupings.................................. 1 . 1 . 5 Various other algebraic buildings besides fields, rings, and groups..... 1 . 2 Isomorphisms, homomorphisms, and so forth...................... 1 . installment payments on your 1 Isomorphisms.............................. 1 . installment payments on your 2 Homomorphisms............................ 1 . installment payments on your 3 Monomorphisms and epimorphisms.................. 1 . installment payments on your 4 Endomorphisms and automorphisms................. 1 . a few A little number theory............................. 1 ) 3. 1 Mathematical debut ? initiation ? inauguration ? introduction on the natural numbers In......... 1 . a few. 2 Divisibility................................ 1 . 3. 3 Perfect numbers............................. 1 ) 3. some The Euclidean algorithm........................ 1 . 4 The essential theorem of arithmetic: the unique factorization theorem.

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2 Fields

2 . you Introduction to fields................... 2 . 1 ) 1 Definition of fields................ 2 . 1 . 2 Subtraction, division, interminables, and powers..

2 . 1 . 3 Real estate that follow from your axioms.....

2 . 1 . 5 Subfields...................... 2 . 1 . your five Polynomials and fields of rational functions...

2 . 1 . 6 Vector spaces above arbitrary fields........

2 . a couple of Cyclic rings and finite fields............... installment payments on your 2 . one particular Equivalence contact...............

2 . installment payments on your 2 The cyclic diamond ring Zn................

installment payments on your 2 . three or more The cyclic prime fields Zp............

2 . 2 . 5 Characteristics of fields, and prime fields....

2 . several Field Plug-ins, algebraic fields, the complex numbers installment payments on your 3. one particular An algebraic field.................

2 . 3. two The field of complex numbers C.........

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iv

CONTENTS

2 . 4

installment payments on your 5

2 . 3. several General quadratic extensions.......

Genuine numbers and ordered fields.........

2 . some. 1 Purchased fields...............

2 . 4. a couple of Archimedean instructions............

2 . 4. 3 Complete ordered fields..........

Skew fields (division rings) and the quaternions


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