Bernard L. Johnston, Fred Richman

Numbers and Symmetry

An Introduction to Algebra

• Broschiertes Buch

Produktbeschreibung

This textbook presents modern algebra from the ground up using numbers and symmetry. The idea of a ring and of a field are introduced in the context of concrete number systems. Groups arise from considering transformations of simple geometric objects. The analysis of symmetry provides the student with a visual introduction to the central algebraic notion of isomorphism. Designed for a typical one-semester undergraduate course in modern algebra, it provides a gentle introduction to the subject by allowing students to see the ideas at work in accessible examples, rather than plunging them immediately into a sea of formalism. The student is involved at once with interesting algebraic structures, such as the Gaussian integers and the various rings of integers modulo n, and is encouraged to take the time to explore and become familiar with those structures. In terms of classical algebraic structures, the text divides roughly into three parts:

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Produktdetails

• Verlag: CRC Press
• Seitenzahl: 272
• Erscheinungstermin: 7. Januar 1997
• Englisch
• Abmessung: 234mm x 156mm x 15mm
• Gewicht: 418g
• ISBN-13: 9780849303012
• ISBN-10: 084930301X
• Artikelnr.: 21787459

Autorenporträt

Department of Mathematical Sciences Florida Atlantic University Boca Raton, Florida. Department of Mathematical Sciences Florida Atlantic University Boca Raton, Florida.

Inhaltsangabe

1 New numbers
1.1 A planeful of integers, Z[i]
1.2 Circular numbers, Zn
1.3 More integers on the number line, Z [V]
1.4 Notes
2 The division algorithm
2.1 Rational integers
2.2 Norms
2.2.1 Gaussian integers
2.2.2 Z[V2]
2.3 Gaussian numbers
2.4 Q (V2)
2.5 Polynomials
2.6 Notes
3 The Euclidean algorithm
3.1 Bezout's equation
3.2 Relatively prime numbers
3.3 Gaussian integers
3.4 Notes.
4 Units
4.1 Elementary properties
4.2 Bezout's equation
4.2.1 Casting out nines
4.3 Wilson's theorem
4.4 Orders of elements: Fermat and Euler
4.5 Quadratic residues
4.6 Z[\ /2)
4.7 Notes
5 Primes
5.1 Prime numbers
5.2 Gaussian primes
5.3 Z [s /2]
5.4 Unique factorization into primes.
5.5 Zn.
5.6 Notes
6 Symmetries
6.1 Symmetries of figures in the plane
6.2 Groups
6.2.1 Permutation groups
6.2.2 Dihedral groups
6.3 The cycle structure of a permutation
6.4 Cyclic groups
6.5 The alternating groups
6.5.1 Even and odd permutations
6.5.2 The sign of a permutation
6.6 Notes
7 Matrices
7.1 Symmetries and coordinates
7.2 Two
by
two matrices
7.3 The ring of matrices
7.4 Units
7.5 Complex numbers and quaternions
7.6 Notes
8 Groups
8.1 Abstract groups
8.2 Subgroups and cosets
8.3 Isomorphism
8.4 The group of units of a finite field
8.5 Products of groups
8.6 The Euclidean groups E(l), E(2) and E(3)
8.7 Notes
9 Wallpaper patterns
9.1 One
dimensional patterns
9.2 Plane lattices
9.3 Frieze patterns
9.4 Space groups
9.5 The 17 plane groups
9.6 Notes
10 Fields
10.1 Polynomials over a field
10.2 Kronecker's construction of simple field extensions
10.2.1 A four
element field, Kron(Z2, X2 + X + 1)
10.2.2 A sixteen
element field, Kron(Z2, X4
f X + 1)
10.3 Finite fields
10.4 Notes
11 Linear algebra
11.1 Vector spaces
11.2 Matrices
11.3 Row space and echelon form
11.4 Inverses and elementary matrices
11.5 Determinants
11.6 Notes
12 Error
correcting codes
12.1 Coding for redundancy
12.2 Linear codes
12.2.1 A Hamming code
12.3 Parity
check matrices
12.4 Cyclic codes
12.5 BCH codes
12.5.1 A two
error
correcting code
12.5.2 Designer codes
12.6 CDs
12.7 Notes
13 Appendix: Induction
13.1 Formulating the n
th statement
13.2 The domino theory: iteration.
13.3 Formulating the induction statement
13.3.1 Summary of steps
13.4 Squares
13.5 Templates
13.6 Recursion
13.7 Notes
14 Appendix: The usual rules
14.1 Rings
14.2 Notes
Index.