Check regularly the following ever-growing list of
ever-changing pdf files
(the files grew out of lectures in various semesters;
hence the occasional overlaps):
some basic algebraic structures
try these simple algebra problems
polynomials, field extensions, Euclidean constructions
spanning sets, independent sets, bases, dimension
Here is an interesting
generalization
of the notion of subspaces, spanning sets, independent sets, and bases.
integers that can be written as the sum of two squares
Wilson's theorem and Fermat's ``little theorem"
additive functions - the Cauchy equation
Some further topics for talks (hopefully with writeups later):
The Bolyai-Gerwien theorem
Hilbert's third problem; Dehn's solution
Cantor's world: Cardinalities; The
Bernstein-Schröder
theorem;
The Banach-Tarski paradox
Lagrange's Four Squares Theorem
Names:
Banach, Stefan (Polish; 1892-1945)
[born in Kraków, then Austria-Hungary now Poland]
Bernstein, Felix (German; 1878-1956)
Bolyai, Farkas (Wolfgang) (Hungarian; 1775-1856)
[father of János Bolyai 1802-1860]
For more names referenced by these pages, see names.htm or the original source: http://www-history.mcs.st-andrews.ac.uk/history/BiogIndex.html