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
