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.

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

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.