Elements of Modern Algebra
1. FUNDAMENTALS. Sets. Mappings. Properties of Composite Mappings (Optional). Binary Operations. Permutations and Inverses. Matrices. Relations. Key Words and Phrases. A Pioneer in Mathematics: Arthur Cayley. 2. THE INTEGERS. Postulates for the Integers (Optional). Mathematical Induction. Divisibility. Prime Factors and Greatest Common Divisor. Congruence of Integers. Congruence Classes. Introduction to Coding Theory (Optional). Introduction to Cryptography (Optional). Key Words and Phrases. A Pioneer in Mathematics: Blaise Pascal. 3. GROUPS. Definition of a Group. Properties of Group Elements. Subgroups. Cyclic Groups. Isomorphisms. Homomorphisms. Key Words and Phrases. A Pioneer in Mathematics: Niels Henrik Abel. 4. MORE ON GROUPS. Finite Permutation Groups. Cayley's Theorem. Permutation Groups in Science and Art (Optional). Cosets of a Subgroups. Normal Subgroups. Quotient Groups. Direct Sums (Optional). Some Results on Finite Abelian Groups (Optional). Key Words and Phrases. A Pioneer in Mathematics: Augustin Louis Cauchy. 5. RINGS, INTEGRAL DOMAINS, AND FIELDS. Definition of a Ring. Integral Domains and Fields. The Field of Quotients of an Integral Domain. Ordered Integral Domains. Key Words and Phrases. A Pioneer in Mathematics: Richard Dedekind. 6. MORE ON RINGS. Ideals and Quotient Rings. Ring Homomorphisms. The Characteristic of a Ring.Maximal Ideals (Optional). Key Words and Phrases. A Pioneer in Mathematics: Amalie Emmy Noether. 7. REAL AND COMPLEX NUMBERS. The Field of Real Numbers. Complex Numbers and Quaternions. De Moivre's Theorem and Roots of Complex Numbers. Key Words and Phrases. A Pioneer in Mathematics: William Rowan Hamilton. 8. POLYNOMIALS. Polynomials over a Ring. Divisibility and Greatest Common Divisor. Factorization in F[x]. Zeros of a Polynomial. Solutions of Cubic and Quartic Equations by Formulas (Optional). Algebraic Extensions of a Field. Key Words and Phrases. A Pioneer in Mathematics: Carl Friedrich Gauss. Appendix: The Basics of Logic.