Numbers and Symmetry-An Introduction to Algebra

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.