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Presents modern algebra from the ground up using numbers and symmetry. This work provides an introduction to the central algebraic notion of isomorphism. Designed for a typical one-semester undergraduate course in modern algebra, it provides an introduction to the subject by allowing students to see the ideas at work in accessible examples.
Presents modern algebra from the ground up using numbers and symmetry. This work provides an introduction to the central algebraic notion of isomorphism. Designed for a typical one-semester undergraduate course in modern algebra, it provides an introduction to the subject by allowing students to see the ideas at work in accessible examples.
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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.
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.
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