Math 223 - Abstract Algebra

Abstract algebraic concepts such as groups, fields, rings, have evolved from
various examples. Abstraction is necessary in order to understand concrete
phenomena, and vice versa. The ideas that seemed abstract yesterday,  seem
more concrete today as they become familiar.

The concept of a group arose from examples of symmetries. The symmetries of
every object or subject of scientific research (e.g. a ball,  a card, a
vector space) satisfies the following axioms. 
(1) The composition  fg  of two symmetries  f  and  g  is a symmetry, and
(fg)h = f(gh) for arbitrary symmetries   f, g and  h.
(2) The identity function   i  defined by  i(x) = x  is a symmetry, and  we
have  i f  = f i  = f  for arbitrary symmetry  f.
(3) The inverse  g  of a symmetry  f   (i.e.  fg = gf = i)  exists, and g is
itself a symmetry.

Group theory has grown up from these 3 axioms.  Example of an immediate
consequence : Apply axiom (1) and prove that  (fg)(uv) = f((gu)v) for
arbitrary symmetries f, g, u  and  v. A harder problem: Let a group  G  have
exactly  3 symmetries; prove that no non-identity symmetry  f  of  G
satisfies  f f   = i .

To solve equation the 2x = 3 one needs rational numbers. All rational numbers
form a field. (This concept is also axiomatic.) The square root of 2 is not a
member of this field, but we can define a larger field F generated by the
square root of 2. The field of real numbers  R is larger than  F, but to be
able to solve all quadratic equations with real coefficients, we need an even
larger field than R. It is the field of complex numbers  C. Every algebraic
equation generates its own field, the splitting field of the equation.

Why are the equations of degrees 2, 3 and 4  solvable in radicals, but the
roots of equation x^5 = 100x + 100  are not expressible in radicals? It turns
out that solvability in radicals depends on the symmetries of the equation's
splitting field. (The group of symmetries for the equation of degree 5 above,
has 120 symmetries. Can you believe it?)

There are numerous applications of  the modern algebra and its axiomatic
method in all branches of mathematics, in physics, and in the practice of
engineering. The background for Math 223 is a standard first course in linear
algebra.