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Abstract Algebra With Applications by Irwin Kra

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Abstract Algebra With Applications written by Irwin Kra . This is an other great mathematics book cover the following topics.

  • Introduction
    Standard Notation and Commonly Used Symbols

  • The integers
    Introduction, Induction, The division algorithm: gcd and lcm, Primes, The rationals, algebraic numbers and other beasts(The rationals, The reals, The complex numbers, The algebraic numbers, The quaternions), Modular arithmetic, Solutions of linear congruences, Euler, Public key cryptography, A collection of beautiful results

  • Foundations
    Naive set theory, Functions, Relations, Order relations on Z and Q (Orders on Z, Orders on Q), The complex numbers

  • Groups
    Permutation groups, The order and sign of a permutation, Definitions and more examples of groups

  • Group homomorphisms and isomorphisms.
    Elementary group theory, Lagrange’s theorem, Homomorphisms, Groups of small order (|G| = 1, |G| = 2, 3, 5, 7 and, in fact, all primes, |G| = 4, |G| = 6, |G| = 8), Homomorphisms and quotients, Isomorphisms (Every group is a subgroup of a permutation group, Solvable groups, MORE sections to be included)

  • Algebraic structures
    A collection of algebraic structures, The algebra of polynomials (The vector space of polynomials of degree n, The Euclidean algorithm (for polynomials), Differentiation, Ideals (Ideals in commutative rings, Ideals in Z and C[x]), CRT revisited, Polynomials over more general fields, Fields of quotients and rings of rational functions

  • Error correcting codes
    ISBN, Groups and codes

  • Roots of polynomials
    Roots of polynomials (Derivatives and multiple roots), Circulant matrices, Roots of polynomials of small degree (Roots of linear and quadratic polynomials, The general case, Roots of cubics, Roots of quartics, Real roots and roots of absolute value 1, What goes wrong for polynomials of higher degree?)

  • Moduli for polynomials
    Polynomials in three guises, An example from high school math: the quadratic polynomial, An equivalence relation, An example all high school math teachers should know: the cubic polynomial, Arbitrary real or complex polynomials, Back to the cubic polynomial, Standard forms for cubics, Solving the cubic, Solving the quartic, Concluding remarks, A moduli (parameter) count

  • Nonsolvability by radicals
    Algebraic extensions of fields, Field embeddings, Splitting fields, Galois extensions, Quadratic, cubic and quartic extensions (Linear extensions, Quadratic extensions, Cubic extensions, Quartic extensions, Nonsolvability)

  • Bibliography

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