Website owner:  James Miller

Groups, Rings, Integral Domains and Fields

Functions, mappings, maps, transformations, operators. Onto, one-to-one, surjective, injective, bijective, identity, product, inverse functions. Group of transformations on a set. Permutation. Symmetric group Sn

Congruence, residue classes of integers modulo m

Products, quotients and roots of complex numbers in polar form, primitive roots of unity

Modern, abstract mathematics

Binary operations

Groups, subgroups, complexes, cosets, transforms, normal subgroups, quotient groups, commutators, composition series, isomorphisms, homomorphisms, automorphisms

Permutations, cyclic permutations (cycles), permutation groups, transpositions. Even and odd permutations. Symmetric and alternating groups. Decomposition of a permutation group into cycles and a product of transpositions.

Cosets

Conjugates, conjugate classes, automorphisms, normal subgroups, quotient groups, homomorphic mapping of groups

Motions of space. Intuitive meaning of the concept of a conjugate transformation of a transformation A

Intuitive interpretation of the commutator of two elements of a group. Commutator subgroups (or derived groups). Commutators of permutations.

Full linear group, Real linear group, Orthogonal group, Affine Group, Euclidean Group

The group of symmetries of the rectangle (the four group)

The Dihedral Group of the Equilateral Triangle

The Dihedral Group of the Square

The symmetric group on four letters, S4

Rings

Ideals, quotient rings, homomorphisms

Integral domains and fields

Axioms satisfied by rings, integral domains and fields

Field. Number field. Extension field. Splitting field. Separable polynomial. Algebraic number. Algebraic element, Transcendental element.

Elementary symmetric polynomials. Viete’s Formulas.

Ground field, splitting field, root field, Galois group, solvable group