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	Product Structures and Division Rings over the Real Numbers

Here's a long summary of answers to questions frequently asked in the sci.math newsgroup concerning products on R^n, especially cross products, and a general review of algebras and (more generally) rings. In particular, we address the question of what division algebras are, and give the answer "1, 2, 4, 8".

These issues are really just part of ring theory ( commutative, associative (noncommutative), or nonassociative) The special role played by the real numbers introduces connections with topology; in particular, the non-existence of division algebras of high dimension was proven using facts about spheres; a FAQ on spheres is a separate page.

This is the division algebra FAQ itself.

I've also saved a few other letters and posts which may be relevant:

• Some history of the "1,2,4,8" theorems.
• The division algebras as composition algebras, allowing product formulas for sums of (1, 2, 4, or 8) squares.
• Citations to doublings, etc., and mention of sphere-packing.
• What about algebras over the algebraic closure of Q?
• Applications to physics [John Baez]
• Applications of octonions in mathematical physics, again [John Baez]
• Citation and pointer for use of octonions in physics
• Why there are no 3-dimensional real fields
• Are there (associative) Banach Algebras which are division rings? (only the finite-dimensional ones).
• Citation: book of Kantor and Solodovnikov on real algebras
• Is there a cross-product in R^n?
• Why cross-products exist only in dimensions 3 and 7
• What are the "cross-products" in dimension n?
• Generalization of DeMoivre's Theorem to N-dimensional spaces
• Copy of the post made announcing availability of the FAQ.