# 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
• 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
• 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