Introduction to abstract algebra

Content: Numbers Ordering numbersThe Well-Ordering PrincipleDivisibilityThe Division AlgorithmGreatest common divisorsThe Euclidean AlgorithmPrimes and irreduciblesThe Fundamental Theorem of ArithmeticExercisesStudy projectsNotesFunctionsSpecifying functionsComposite functionsLinear functionsSemigroups of functionsInjectivity and surjectivityIsomorphismsGroups of permutationsExercisesStudy projectsNotesSummaryEquivalenceKernel and equivalence relationsEquivalence classesRational numbersThe First Isomorphism Theorem for SetsModular arithmeticExercisesStudy projectsNotesGroups and Monoids SemigroupsMonoidsGroupsComponentwise structurePowersSubmonoids and subgroups CosetsMultiplication tablesExercisesStudy projectsNotesHomomorphisms HomomorphismsNormal subgroupsQuotientsThe First Isomorphism Theorem for GroupsThe Law of ExponentsCayley's TheoremExercisesStudy projectsNotesRings RingsDistributivitySubringsRing homomorphismsIdealsQuotient ringsPolynomial ringsSubstitutionExercisesStudy projectsNotesFields Integral domainsDegreesFieldsPolynomials over fieldsPrincipal ideal domains Irreducible polynomialsLagrange interpolationFields of fractionsExercisesStudy projectsNotesFactorizationFactorization in integral domainsNoetherian domainsUnique factorization domainsRoots of polynomialsSplitting fieldsUniqueness of splitting fieldsStructure of finite fieldsGalois fieldsExercisesStudy projects NotesModulesEndomorphismsRepresenting a ring ModulesSubmodulesDirect sumsFree modulesVector spacesAbelian groupsExercisesStudy projectsNotesGroup ActionsActionsOrbitsTransitive actionsFixed pointsFaithful actionsCoresAlternating groupsSylow TheoremsExercisesStudy projects NotesQuasigroupsQuasigroupsLatin squaresDivisionQuasigroup homomorphismsQuasigroup homotopiesPrincipal isotopyLoopsExercisesStudy projectsNoteIndex