Learning Abstract Algebra with ISETL

Learning Abstract Algebra with ISETL

Learning Abstract Algebra with ISETL by Ed Dubinsky and Uri Leron

Contents of Learning Abstract Algebra with ISETL

  • Mathematical Constructions in ISETL
  • Using ISETL
  • Activities
  • Getting started
  • Simple objects and operations on them
  • Control statements
  • Exercises
  • Compound objects and operations on them
  • Activities
  • TUples
  • Sets
  • Set and tuple formers
  • Set operations
  • Permutations
  • Quantification
  • Miscellaneous ISETL features
  • VISETL
  • Exercises
  • Functions in ISETL
  • Activities
  • Funcs
  • Alternative syntax for funcs
  • Using funcs to represent situations
  • Funcs for binary operations
  • Funcs to test properties
  • Smaps
  • Procs
  • Exercises
  • Groups
  • Getting acquainted with groups
  • Activities
  • Definition of a group
  • Examples of groups
  • Number systems
  • Integers mod n
  • Symmetric groups
  • Symmetries of the square
  • Groups of matrices
  • Elementary properties of groups
  • Exercises
  • The modular groups and the symmetric groups
  • Activities
  • The modular groups Zn
  • The symmetric groups Sn
  • Orbits and cycles
  • Exercises
  • Properties of groups
  • Activities
  • The specific and the general
  • The cancellation law-An illustration of the
  • abstract method
  • How many groups are there?
  • Classifying groups of order
  • Looking ahead-subgroups
  • Summary of examples and non-examples of groups
  • Exercises
  • Subgroups
  • Definitions and examples
  • Activities
  • Subsets of a group
  • Definition of a subgroup
  • Examples of subgroups
  • Embedding one group in another
  • Conjugates
  • Cycle decomposition and conjugates in Sn
  • Exercises
  • Cyclic groups and their subgroups
  • Activities
  • The subgroup generated by a single element
  • Cyclic groups
  • The idea of the proof
  • Generators
  • Generators of Sn
  • Parity-even and odd permutations
  • Determining the parity of a permutation
  • Exercises
  • Lagrange’s theorem
  • Activities
  • What Lagrange’s theorem is all about Cosets
  • The proof of Lagrange’s theorem
  • Exercises
  • The Fundamental Homomorphism Theorem
  • Quotient groups
  • Activities
  • Normal subgroups
  • Multiplying cosets by representatives
  • The quotient group
  • Exercises
  • Homomorphisms
  • Activities
  • Homomorphisms and kernels
  • Examples
  • Invariants
  • Homomorphisms and normal subgroups
  • An interesting example
  • Isomorphisms
  • Identifications
  • Exercises
  • The homomorphism theorem
  • Activities
  • The canonical homomorphism
  • The fundamental homomorphism theorem
  • Exercises
  • Rings
  • Ideals
  • Activities
  • Definition of a ring
  • Examples of rings
  • Rings with additional properties
  • Integral domains
  • Fields
  • Constructing new rings from old-matrices
  • Constructing new rings from old-polynomials
  • Constructing new rings from old-functions
  • Elementary properties-arithmetic
  • Exercises
  • Activities
  • Analogies between groups and rings
  • Subrings
  • Definition of subring
  • Examples of subrings
  • Subrings of Zn and Z
  • Subrings of M (R)
  • Subrings of polynomial rings
  • Subrings of rings of functions
  • Ideals and quotient rings
  • Definition of ideal
  • Examples of ideals
  • Elementary properties of ideals
  • Elementary properties of quotient rings
  • Quotient rings that are integral domainsprime ideals
  • Quotient rings that are fields-maximal ideals
  • Exercises
  • Homomorphisms and isomorphisms
  • Activities
  • Definition of homomorphism and isomorphism
  • Group homomorphisms vs ring homomorphisms
  • Examples of homomorphisms and isomorphisms
  • Homomorphisms from Zn to Zk
  • Homomorphisms of Z
  • Homomorphisms of polynomial rings
  • Embeddings-Z, Zn as universal subobjects
  • The characteristic of an integral domain and a field
  • Properties of homorphisms
  • Preservation
  • Ideals and kernels of ring homomorphisms
  • The fundamental homomorphism theorem
  • The canonical homomorphism
  • The fundamental theorem
  • Homomorphic images of Z, Zn
  • Identification of quotient rings
  • Exercises
  • Factorization in Integral Domains
  • Divisibility properties of integers and polynomials
  • Activities
  • The integral domains Z, Q[x]
  • Arithmetic and factoring
  • The meaning of unique factorization
  • Arithmetic of polynomials
  • Long division of polynomials
  • Division with remainder
  • Greatest Common Divisors and the Euclidean
  • algorithm
  • Exercises
  • Euclidean domains and unique factorization
  • Activities
  • Gaussian integers
  • Can unique factorization fail?
  • Elementary properties of integral domains
  • Euclidean domains
  • Examples of Euclidean domains
  • Unique factorization in Euclidean domains
  • Exercises
  • The ring of polynomials over a field
  • Unique factorization in F[x]
  • Roots of polynomials
  • The evaluation homomorphism
  • Reducible and irreducible polynomials
  • Examples
  • Extension fields
  • Construction of the complex numbers
  • Splitting fields
  • Exercises

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