# CONSTRUCTIVE ALGEBRA

## Section 1, 11:00-11:50 MWF, SE 215

• ### Notes from 2008

• Henri Lombardi's constructive mathematics page. Henri is one of the foremost constructive algebraists in the world. There are lots of good links here, including some in English!

• Here is a review (in English) of the book on constructive algebra by Lombardi and Quitté.

• Take a look at Douglas's FAQ. Douglas may be the best known constructive mathematician in the world. At least for the variety of constructive mathematics that I do.

• First chapter of the textbook, html version, in case you need it at first

Constructive algebra goes back to the work of Kronecker and Brouwer. It is traditional abstract algebra thought of algorithmically, but not necessarily done algorithmically. It's not computer algebra or computational algebra. The text is "A course in constructive algebra" by Ray Mines, Fred Richman, and Wim Ruitenburg, Springer, 1988. Here is what it says on the back of the book:
In this book, the authors present the fundamental structures of modern algebra from a constructive point of view. Beginning with basic notions, the authors proceed to treat PIDs, field theory (including Galois theory), factorization of polynomials, Noetherian rings, valuation theory, and Dedekind domains.
Linear algebra is also treated in the book. We won't be able to cover everything. Moreover, there is no need to cover everything. The point of this course is to introduce the basic ideas and techniques of constructive algebra. You can then apply these to any branch of algebra (or of mathematics) that strikes your fancy.

I will tailor the topics to the audience. There is no advanced algebra requirement. It will essentially be a level playing field because no one will know how to play this game.

Constructive algebra is algebra. There is a shift of point of view that makes us fussier as to what constitutes a proof of existence, and that makes us favor direct proofs over indirect ones, but the subject matter is the same. However, there is a bit of a culture shock involved. The existence claim that every nonempty set of positive integers contains a least element is rejected. The corresponding algebraic statement that every ideal in the ring of integers is principal is also rejected. It turns out that you can do the kind of mathematics that you are familiar with perfectly well without these tools. What do you gain? The usual fruits of generalization: greater insight, richer structures, more powerful theorems. Every theorem in constructive algebra is a theorem in classical algebra, but because there is a higher standard of proof, these theorems are valid in greater generality. In particular, they can be interpreted computationally.

And, it's a barrel of laughs.

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