Learning Abstract Algebra with ISETL

1 Mathematical Constructions in ISETL.- 1.1 Using ISETL.- 1.1.1 Activities.- 1.1.2 Getting started.- 1.1.3 Simple objects and operations on them.- 1.1.4 Control statements.- 1.1.5 Exercises.- 1.2 Compound objects and operations on them.- 1.2.1 Activities.- 1.2.2 Tuples.- 1.2.3 Sets.- 1.2.4 Set and tuple formers.- 1.2.5 Set operations.- 1.2.6 Permutations.- 1.2.7 Quantification.- 1.2.8 Miscellaneous ISETL features.- 1.2.9 VISETL.- 1.2.10 Exercises.- 1.3 Functions in ISETL.- 1.3.1 Activities.- 1.3.2 Funcs.- 1.3.3 Alternative syntax for funcs.- 1.3.4 Using funcs to represent situations.- 1.3.5 Funcs for binary operations.- 1.3.6 Funcs to test properties.- 1.3.7 Smaps.- 1.3.8 Procs.- 1.3.9 Exercises.- 2 Groups.- 2.1 Getting acquainted with groups.- 2.1.1 Activities.- 2.1.2 Definition of a group.- 2.1.3 Examples of groups.- Number systems.- Integers mod n.- Symmetric groups.- Symmetries of the square.- Groups of matrices.- 2.1.4 Elementary properties of groups.- 2.1.5 Exercises.- 2.2 The modular groups and the symmetric groups.- 2.2.1 Activities.- 2.2.2 The modular groups Zn.- 2.2.3 The symmetric groups Sn.- Orbits and cycles.- 2.2.4 Exercises.- 2.3 Properties of groups.- 2.3.1 Activities.- 2.3.2 The specific and the general.- 2.3.3 The cancellation law—An illustration of the abstract method.- 2.3.4 How many groups are there?.- Classifying groups of order 4.- 2.3.5 Looking ahead—subgroups.- 2.3.6 Summary of examples and non-examples of groups.- 2.3.7 Exercises.- 3 Subgroups.- 3.1 Definitions and examples.- 3.1.1 Activities.- 3.1.2 Subsets of a group.- Definition of a subgroup.- 3.1.3 Examples of subgroups.- Embedding one group in another.- Conjugates.- Cycle decomposition and conjugates in Sn.- 3.1.4 Exercises.- 3.2 Cyclic groups and their subgroups.- 3.2.1 Activities.- 3.2.2 The subgroup generated by a single element.- 3.2.3 Cyclic groups.- The idea of the proof.- 3.2.4 Generators.- Generators of Sn.- Parity—even and odd permutations.- Determining the parity of a permutation.- 3.2.5 Exercises.- 3.3 Lagrange’s theorem.- 3.3.1 Activities.- 3.3.2 What Lagrange’s theorem is all about.- 3.3.3 Cosets.- 3.3.4 The proof of Lagrange’s theorem.- 3.3.5 Exercises.- 4 The Fundamental Homomorphism Theorem.- 4.1 Quotient groups.- 4.1.1 Activities.- 4.1.2 Normal subgroups.- Multiplying cosets by representatives.- 4.1.3 The quotient group.- 4.1.4 Exercises.- 4.2 Homomorphisms.- 4.2.1 Activities.- 4.2.2 Homomorphisms and kernels.- 4.2.3 Examples.- 4.2.4 Invariants.- 4.2.5 Homomorphisms and normal subgroups.- An interesting example.- 4.2.6 Isomorphisms.- 4.2.7 Identifications.- 4.2.8 Exercises.- 4.3 The homomorphism theorem.- 4.3.1 Activities.- 4.3.2 The canonical homomorphism.- 4.3.3 The fundamental homomorphism theorem.- 4.3.4 Exercises.- 5 Rings.- 5.1 Rings.- 5.1.1 Activities.- 5.1.2 Definition of a ring.- 5.1.3 Examples of rings.- 5.1.4 Rings with additional properties.- Integral domains.- Fields.- 5.1.5 Constructing new rings from old—matrices.- 5.1.6 Constructing new rings from old—polynomials.- 5.1.7 Constructing new rings from old—functions.- 5.1.8 Elementary properties—arithmetic.- 5.1.9 Exercises.- 5.2 Ideals.- 5.2.1 Activities.- 5.2.2 Analogies between groups and rings.- 5.2.3 Subrings.- Definition of subring.- 5.2.4 Examples of subrings.- Subrings of Zn and Z.- Subrings of M(R).- Subrings of polynomial rings.- Subrings of rings of functions.- 5.2.5 Ideals and quotient rings.- Definition of ideal.- Examples of ideals.- 5.2.6 Elementary properties of ideals.- 5.2.7 Elementary properties of quotient rings.- Quotient rings that are integral domains—
prime ideals.- Quotient rings that are fields—maximal ideals.- 5.2.8 Exercises.- 5.3 Homomorphisms and isomorphisms.- 5.3.1 Activities.- 5.3.2 Definition of homomorphism and isomorphism.- Group homomorphisms vs. ring homomorphisms.- 5.3.3 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.- 5.3.4 Properties of homorphisms.- Preservation.- Ideals and kernels of ring homomorphisms.- 5.3.5 The fundamental homomorphism theorem.- The canonical homomorphism.- The fundamental theorem.- Homomorphic images of Z, Zn.- Identification of quotient rings.- 5.3.6 Exercises.- 6 Factorization in Integral Domains.- 6.1 Divisibility properties of integers and polynomials.- 6.1.1 Activities.- 6.1.2 The integral domains Z, Q[x].- Arithmetic and factoring.- The meaning of unique factorization.- 6.1.3 Arithmetic of polynomials.- Long division of polynomials.- 6.1.4 Division with remainder.- 6.1.5 Greatest Common Divisors and the Euclidean algorithm.- 6.1.6 Exercises.- 6.2 Euclidean domains and unique factorization.- 6.2.1 Activities.- 6.2.2 Gaussian integers.- 6.2.3 Can unique factorization fail?.- 6.2.4 Elementary properties of integral domains.- 6.2.5 Euclidean domains.- Examples of Euclidean domains.- 6.2.6 Unique factorization in Euclidean domains.- 6.2.7 Exercises.- 6.3 The ring of polynomials over a field.- 6.3.1 Unique factorization in F[x].- 6.3.2 Roots of polynomials.- 6.3.3 The evaluation homomorphism.- 6.3.4 Reducible and irreducible polynomials.- Examples.- 6.3.5 Extension fields.- Construction of the complex numbers.- 6.3.6 Splitting fields.- 6.3.7 Exercises.