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Problem Solving by George Polya

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Problem Solving written by George Polya . This is an other great mathematics book cover the following topics of Problem Solving.

  • Introduction
    What is Abstract Algebra? (History, Abstraction), Motivating Examples (The Integers, Matrices), The integers mod n (The Euclidean Algorithm)

  • Group Theory
    Definitions and Examples of Groups (Binary Operations, Groups, Group Tables, Remarks on Notation), The Symmetric and Dihedral Groups (The Symmetric Group, The Dihedral Group), Basic Properties of Groups, The Order of an Element and Cyclic Groups (Cyclic Groups, Classification of Cyclic Groups), Subgroups (Cyclic Subgroups, Subgroup Criteria, Subgroups of Cylic Groups), Lagrange’s Theorem (Equivalence Relations, Cosets), Homomorphisms (Basic Properties of Homomorphisms), The Symmetric Group Redux (Cycle Decomposition, Application to Dihedral Groups, Cayley’s Theorem, Even and Odd Permutations and the Alternating Group), Kernels of Homomorphisms, Quotient Groups and Normal Subgroups (The Integers mod n, General Quotient Groups, Normal Subgroups, The First Isomorphism Theorem, Aside: Applications of Quotient Groups), Direct Products of Groups, The Classification of Finite Abelian Groups

  • Ring Theory
    Rings, Basic Facts and Properties of Rings (The Quaternions), Ring Homomorphisms and Ideals, Quotient Rings, Polynomials and Galois Theory, Act I: Roots of Polynomials, Act II: Field Extensions, Act III: Galois Theory (Epilogue)

  • Set Theory
    Sets, Constructions on Sets, Set Functions, Notation

  • Techniques for Proof Writing
    Basic Proof Writing, Proof by Contradiction, Mathematical Induction, Proof by Contrapositive, Tips and Tricks for Proofs

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