New for: D1, D2, D3, D4, D5
Despite the title, this will not be a general introduction to algebra, geometry, or algebraic geometry. In the short time we have, I'd rather like to address three items which I consider to be particularly interesting:
1) Rational, real, complex, algebraic, and real algebraic numbers.
Computing with algebraic numbers.
2) The first-order theory of real closed fields is decidable.
3) Gröbner bases. Affine rings. Multiplication as a linear map.
Rational Univariate Representation.
Please see http://www.mpi-sb.mpg.de/~arno/minicourse04/ for more.