Changes

February 5, 2011

We have received a new set of donations, again from a live-TeXed course. One thing that I hadn’t seen before is a theorem of McCoy that gives a criterion (in the “density” of certain ideals generated by minors) of when a matrix mapping of free modules is injective.

In addition, various edits keep being made, slowly. There have been several articles in the Monthly that give short semi-expository pieces of mathematics; a brief account of some of them have appeared in the book. Here is one, giving a short proof of the following lemma: a field extension which is finitely generated as an algebra is algebraic.

To see this, one can use the following argument of McCabe. Let $k'/k$ be an extension finitely generated as an algebra. There is a subring $k[x_1, \dots, x_n]$ over which $k’$ is algebraic. The finite generation hypothesis implies that we can find a localization $k[x_1, \dots, x_n]_y$ over which $k’$ is even integral. But if one has an integral extension and the top ring is a field, then so is the bottom one; this implies $k[x_1, \dots,x_n]_y$ is a field. So $y \in k[x_1, \dots, x_n]$ must lie in every maximal ideal, meaning it is zero, contradiction.