In Mathematics 504, we will study some algebraic concepts that arise in finding roots of polynomials in one indeterminate. We will study field extensions, algebraic extensions, splitting fields, and algebraic closure. Next we will consider the expression of the coefficients of a polynomial in terms of its roots. This will lead into the study of the symmetric group and symmetric polynomials, from which Galois theory is a natural step. In Galois theory, a deeper study is made of permutations of roots, field extensions, and group theory. Next up will be some linear algebra and module theory over rings. The development of determinant theory will allow us to say more about polynomials and their discriminants.

In Mathematics 505, we will study bilinear forms and the orthogonal groups O(n). We will then study the finite subgroups of O(2) and O(3). Maybe we'll look briefly at finite subgroups of O(4) as well. This is intimately connected with the study of regular polygons, regular polyhedra, and regular polytopes in four dimensions. Some of these groups will be studied further during a brief look at representation theory. Then we'll begin the study of zeroes of polynomials in several indeterminates by looking at hypersurfaces - the loci of zeroes of single polynomials.

In Mathematics 506, we'll study varieties, the loci of zeroes of families of polynomials in several indeterminates. The relationship between varieties and their rings of regular functions will be emphasized, with a detailed look at various versions of the Nullstellensatz. Dimension theory of varieties will also be developed, via transcendence degree and Krull dimension. The notion of integrality will play a central role. If time permits, we'll take a short look at invariant theory, culminating in a look at the invariant theory of the finite subgroups of O(2) and O(3).