Math 58R Valuation Theory

Math 58R is a course on valuation theory.

The absolute value on Q has the properties: (i) |x| ≥ 0, with |x| = 0 if and only if x = 0; (ii) |xy| = |x| |y|; (iii) |x + y| ≤ |x| + |y|. There are other functions with the same properties, and they are called valuations. The simultaneous study of all valuations on a field, such as Q, leads to an understanding of Q that is not possible by regarding Q as a subset, say, of R.

In math 58R, we will study valuations on a field, the topologies that they induce, prolongations of valuations in field extensions, and we will use valuations to gain profound insight into algebraic numbers.

The course in addressed to graduate students of mathematics and well-prepared senior students. Acquaintance with metric spaces is useful, but not indispensible.