(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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In particular, an emphasis is \ made to create an environment in which the packages can also be suitable for \ pedagogical purposes. To this end, many functions have a ", StyleBox["Textual", FontFamily->"Courier"], " mode (providing information about the function call) and a ", StyleBox["Visual", FontFamily->"Courier"], " mode (providing a graphic to illustrate a concept), in addition to the \ default ", StyleBox["Computational", FontFamily->"Courier"], " mode." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "What is ", StyleBox["Exploring Abstract Algebra with Mathematica", FontSlant->"Italic"], "?" }], "Section"], Cell[TextData[{ "\nThe packages in ", StyleBox["AbstractAlgebra", FontFamily->"Courier"], " form the foundation for a series of 14 group labs and 13 ring labs \ designed to help students conceptualize abstract algebra. These are combined \ with documentation for ", StyleBox["AbstractAlgebra", FontFamily->"Courier"], " in a book entitled ", StyleBox["Exploring Abstract Algebra with Mathematica", FontSlant->"Italic"], " (EAAM) published by Springer Verlag (fall/winter 1998)." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Who are the authors?", "Section"], Cell[TextData[{ "Al Hibbard (Central College)\n", ButtonBox["hibbarda@central.edu ", ButtonData:>{ URL[ "mailto://hibbarda@central.edu "], None}, ButtonStyle->"Hyperlink"], "\n", ButtonBox["http://www.central.edu/homepages/hibbarda/hibbard.html", ButtonData:>{ URL[ "http://www.central.edu/homepages/hibbarda/hibbard.html"], None}, ButtonStyle->"Hyperlink"], "\n\nKen Levasseur (UMass-Lowell)\n", ButtonBox["Levasseuk@woods.uml.edu", ButtonData:>{ URL[ "mailto:Levasseuk@woods.uml.edu"], None}, ButtonStyle->"Hyperlink"], "\n", ButtonBox["http://www.uml.edu/Dept/Math/LevasseuK.html", ButtonData:>{ URL[ "http://www.uml.edu/Dept/Math/LevasseuK.html"], None}, ButtonStyle->"Hyperlink"], "\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Summary of packages' capabilities", "Section"], Cell[TextData[{ "Currently, the packages are capable of handling many of the types of \ objects encountered in a first-year undergraduate abstract algebra course. \ This includes working with (finite) groups, rings, fields, and morphisms and \ functions related to each of these objects. There are a large number of \ built-in groups (including such standard groups as ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_n\)]], ", ", Cell[BoxData[ \(TraditionalForm\`U\_n\)]], " (units of ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_n\)]], "), ", Cell[BoxData[ \(TraditionalForm\`S\_n\)]], ", ", Cell[BoxData[ \(TraditionalForm\`D\_n\)]], ", as well as direct products and quotients of these) and rings (including \ ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_n\)]], ", Boolean rings and Lattice rings, as well as polynomial, matrix and \ function extension rings). One can also create functions (Morphoids) between \ groups or rings and investigate if these are morphisms.\n\nBelow is an \ abbreviated list of the functions in these packages (with many of the names \ being self-documenting as to the functionality)." }], "Text"], Cell["\<\ AbelianQ, Adjoin, Alternating, Annihilator, Associates, \ AssociativeQ, Automorphism, BooleanRing, CartesianProduct, CayleyTable, \ Centralizer, Characteristic, ClosedQ, Closure, Commutators, \ CommutatorSubgroup, ConjugacyClass, Cosets, Cyclic, CyclicGenerators, \ CyclicQ, DiagonalMatrices, Dihedral, DirectProduct, DisjointCyclesQ, \ ElementQ, Elements, ElementToPower, EvenPermutationQ, ExtensionDegree, \ FieldQ, FormGroupoid, FormGroupoidByTable, FormGroupoidFromCycles, \ FormMorphoid, FormRingoid, FormRingoidByTable, FromCycles, GaussianIntegerQ, \ GeneralLinear, GenerateGroupoid, GF, GroupCenter, GroupExponent, \ GroupIdentity, HasIdentityQ, HasInversesQ, HasZeroQ, HermitianQ, \ HomomorphismQ, IdealQ, Idempotents, InducedIsomorphism, InjectiveQ, \ InnerAutomorphism, InnerAutomorphismGroup, Inverses, IsomorphismQ, Kernel, \ Klein4, LeftIdentity, LeftInverse, ModpIrreducibilityQ, MorphismQ, Morphoid, \ MorphoidComposition, MultiplicativeGroupoid, MultiplyCycles, \ MultiplyPermutations, NegationOf, NilpotentQ, Nilpotents, \ NonAssociatingTriples, NonCommutingPairs, Normalizer, NormalQ, Orbit, \ OrderOfElement, Orders, Parity, PermutationInverse, Poly, \ PolynomialEvaluation, PolynomialsUpToDegreeN, PrimeIdealQ, \ PrimitivePolynomials, PrincipalIdeal, ProperSubsetQ, QuaternionGroup, \ QuotientGroup, QuotientRing, RandomElement, RandomElements, RandomMatrix, \ RandomPermutation, RightCoset, RightCosets, RootsOfUnity, SemiGroupQ, \ SkewSymmetricQ, SpecialLinear, Stabilizer, SubgroupGenerated, SubgroupQ, \ SubringQ, SubsetQ, SurjectiveQ, Symmetric, TableOfPowers, ToCycles, \ ToOrdinaryPolynomial, ToTranspositions, TwistedZ, U, UnitQ, Units, Visual2, \ VisualizeMorphoid, WithUnityQ, Z, ZdNorm, ZeroDivisorQ, ZeroDivisors, Zeros\ \ \>", "Text"], Cell["\<\ In addition, the following standard functions (including others) \ have been extended for working with matrices and polynomials over general \ rings:\ \>", "Text"], Cell["\<\ Det, Dot, MatrixPower, PolynomialDivision, PolynomialGCD, \ PolynomialLCM, PolynomialQuotient, PolynomialRemainder, Solve\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Titles of labs in EAAM", "Section"], Cell[TextData[{ "Group labs:\n1. ", StyleBox["Using symmetry to uncover a group,", FontSlant->"Italic"], " 2. ", StyleBox["Determining the symmetry group of a given figure", FontSlant->"Italic"], ", 3. ", StyleBox["Is this a group?", FontSlant->"Italic"], ", 4. ", StyleBox["Let's get these orders straight!", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[",", Evaluatable->False, AspectRatioFixed->True], " 5. ", StyleBox["Subversively grouping our elements", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], "6. ", StyleBox["Cycling through the groups,", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], " 7. ", StyleBox["Permutations,", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], " 8. ", StyleBox["Isomorphisms", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], ", 9. ", StyleBox["Automorphisms", FontSlant->"Italic"], ", 10. ", StyleBox["Direct Products", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], ", 11. ", StyleBox["Cosets", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], ", 12. ", StyleBox["Normality and Factor groups", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], ", 13. ", StyleBox["Homomorphisms", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], ", 14: ", StyleBox["Rotational groups of regular polyhedra", FontSlant->"Italic"], "\n\nRing labs:\n1. ", StyleBox["An Introduction to Ringoids and Rings", FontSlant->"Italic"], ", 2. ", StyleBox["An Introduction to Rings: part two", FontSlant->"Italic"], ", 3. ", StyleBox["An ideal part of rings", FontSlant->"Italic"], ", 4. ", StyleBox["What does ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm \`\[DoubleStruckCapitalZ][i]/ \[LeftAngleBracket]a + b\ i\[RightAngleBracket]\)]], StyleBox[" look like?", FontSlant->"Italic"], ", 5. ", StyleBox["Ring homomorphisms", FontSlant->"Italic"], ", 6. ", StyleBox["Polynomial rings", FontSlant->"Italic"], ", 7. ", StyleBox["Factoring and irreducibility", FontSlant->"Italic"], ", 8. ", StyleBox["Roots of unity", FontSlant->"Italic"], ", 9. ", StyleBox["Cyclotomic polynomials", FontSlant->"Italic"], ", 10. ", StyleBox["Quotient rings of polynomials", FontSlant->"Italic"], ", 11. ", StyleBox["Quadratic field extensions", FontSlant->"Italic"], ", 12. ", StyleBox["Factoring in", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[DoubleStruckCapitalZ][\@d\), "TraditionalForm"], "]"}], TraditionalForm]]], ", 13. ", StyleBox["Finite Fields", FontSlant->"Italic"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Want more information?", "Section"], Cell[TextData[{ "If you go to the EAAM homepage (", ButtonBox["http://www.central.edu/eaam.html", ButtonData:>{ URL[ "http://www.central.edu/eaam.html"], None}, ButtonStyle->"Hyperlink"], "), you will find additional information (including the ", StyleBox["Mathematica", FontSlant->"Italic"], " packages, sample labs, and additional illustrations." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 832}, {0, 604}}, WindowSize->{705, 435}, WindowMargins->{{16, Automatic}, {Automatic, 33}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, StyleDefinitions -> "Report.nb", MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002mT8o?mooh<" ] (*********************************************************************** Cached data follows. 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