Mathematics 5285-6 Honors (Fundamental Structures of Algebra I and II)
Description for prospective students
This is a junior-senior level one-year sequence
in abstract algebra, or what is often called
a "groups, rings, and fields" course.
It is intended for students who intended to go on
to study more abstract mathematics, and who will either
need or enjoy a solid understanding
of algebra at a rigorous level. The expectations of
mathematical maturity and ability to handle rigor
are about as high it gets in our math undergraduate classes,
on a par with the Honors Analysis sequence Math 5615-6.
The exposition often involves lots of
"definition-example-theorem-proof".
Topics covered in the class often include
- linear algebra
(vector spaces, determinants, and possibly Jordan canonical form),
- group theory (including subgroups, normal subgroups, quotient groups,
Sylow's theorems, permutation groups, finite abelian groups, matrix groups)
- ring theory
(ideals, quotient rings, principal ideal domains, factorization,
modular arithmetic, polynomial rings)
- field theory
(finite, algebraic, and transcendental extensions,
extension degree, compass/straightedge constructions,
solvability by radicals, finite fields)
CAUTION: It is not wise to jump into the
second-semester Math 5286 without having just taken Math 5285,
even if one has taken a single semester of a
"groups, and rings, and fields" course before-- the content
in such a single semester can vary greatly.
In such a situation, it is best to consult the instructor
about whether one has the necessary background.
Note that the Math 5285-6 sequence fulfills a math major's
algebra requirement (part of "Column X") ) as described
here .
To suggest some flavor of Math 5285-6, a student might learn ...
- how the 5 Platonic solids relate to the classification
of finite groups of rotations in 3-dimensional space
- why a finite group whose size is divisible by some power pd
of a prime p will always have at least one subgroup of size pd
- how to tell, relatively quickly, whether or not two square matrices
A,B are similar, meaning that B = PAP-1 for some invertible
matrix P
- what all finite fields look like, and how they are constructed
- why it is ultimately impossible, using only a compass and straightedge,
to do the 3 things that frustrated the ancient Greeks:
-
to trisect an arbitrary angle,
-
to "duplicate" a cube,
-
to "square" a circle
-
why the quadratic formula, which does have a generalization
solving 3rd and 4th degree polynomial equations, will never
have a generalization for 5th degree equations.
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