Ebook: Geometric Algebra (Tracts in Pure & Applied Mathematics)

This is a well written classic text on the connection between algebra and geometry. Some of the topics covered include a reconstruction of affine geometry of a field (or division ring) from geometric axioms. A geometry is a triple of a set of points, a set of lines, and a binary relation describing when a point lies on a line and satisfying the following three axioms: (1) any two distinct points are connected by a unique line, (2) given a point P and a line l, there exists a unique line m parallel to l through P, and (3) there exist three points that are not collinear. In order to construct a ring out of this geometry, one must introduce a additional axiom. Axiom 4a: Given any two distinct points P and Q, there exists a translation taking P to Q. In order to show that this ring is a division ring, one must further assume Axiom 4b: Between any two translations with the same direction, there exists a direction preserving homomorphism of the group of translations taking the first translation to the second. Of courses, the notion of translation and direction-preserving needs to be made more precise and this is done in the text.

Of course, the description above covers only a small portion of the book.