An undergraduate course in Abstract Algebra by Robert Howlett

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An undergraduate course in Abstract Algebra written by Robert Howlett
. This is an other great mathematics book cover the following topics of Problem Solving.

0. Prerequisites Concerning notation, Concerning functions, Concerning vector spaces, Some very obvious things about proofs

1. Ruler and compass constructions Three problems, Some examples of constructions, Constructible numbers

2. Introduction to rings Operations on sets, The basic definitions, Two ways of forming rings, Trivial properties of rings

3. The integers Two basic properties of the integers, The greatest common divisor of two integers, Factorization into primes

4. Quotients of the ring of integers Equivalence relations, Congruence relations on the integers, The ring of integers modulo n, Properties of the ring of integers modulo n

5. Some Ring Theory Subrings and subfields, Homomorphisms, Ideals, The characteristic of a ring

6. Polynomials Definitions, Addition and multiplication of polynomials, Constant polynomials, Polynomial functions, Evaluation homomorphisms, The division algorithm for polynomials over a field, The Euclidean Algorithm, Irreducible polynomials, Some examples, Factorization of polynomials, Irreducibility over the rationals

7. More Ring Theory More on homomorphisms, More on ideals, Congruence modulo an ideal, Quotient rings, The Fundamental Homomorphism Theorem

8. Field Extensions Ideals in polynomial rings, Quotient rings of polynomial rings, Fields as quotient rings of polynomial rings, Field extensions and vector spaces, Extensions of extensions, Algebraic and transcendental elements, Ruler and compass constructions revisited, Finite fields