## Introduction to AlgebraThis book is an undergraduate textbook on abstract algebra, beginning with the theories of rings and groups. As this is the first really abstract material students need, the pace here is gentle, and the basic concepts of subring, homomorphism, ideal, etc are developed in detail. Later, as students gain confidence with abstractions, they are led to further developments in group and ring theory (simple groups and extensions, Noetherian rings, and outline of universal algebra, lattices andcategories) and to applications such as Galois theory and coding theory. There is also a chapter outlining the construction of the number systems from scratch and proving in three different ways that trascendental numbers exist. |

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abelian group addition and multiplication algebraic algorithm associative law automorphism axioms binary canonical form closure codewords coefficients column commutative ring complex numbers conjugacy classes construction contains corresponding coset cycle cyclic group define Definition Let denote det(A divides elementary elements of G entry equal equation equivalence classes equivalence relation Euclidean domain example extension factor ring factorisation field F follows function Galois greatest common divisor group G group of order hence homomorphism i?-module induction integral domain inverse irreducible polynomial Isomorphism Theorem kernel lattice Let F Let G linearly independent matrix minimal polynomial module morphisms natural numbers non-zero element normal subgroup notation obtain operation permutation positive integer prime properties Proposition Prove rational numbers real numbers right cosets ring with identity root of unity satisfies Section splitting field subgroup of G submodule subring subset subspace Suppose symmetric theory unique vector space words zero