TITLE: Field structure for R^n
QUESTION [5 upvotes]: Hi!
Is it possible to define a product on R^n for n>2 such that R^n can be made into a field?
R is a field in its own right with the standard operations and R^2 can be made into a field by introducing the product (a,b)*(c,d) = (ac-bd,ad+bc) i.e. the product of the complex numbers.

REPLY [3 votes]: No if you use the usual additive structure on $\mathbb{R}^{n}$ for the field addition; but if you give up commutativity of multiplication, you have the skew-field of hamiltonians, or quaternions, on $\mathbb{R}^{4}$, and if you then give up associativity of multiplication, you have the non-associative Cayley algebra, or octonians , on $\mathbb{R}^{8}$.  The Cayley-Dickson process builds the complex field from the real field, the skew-field of hamiltonians from the complex field, the non-associative Cayley algebra from the hamiltonians, and in general a $2^{n+1}$-dimensional involutive Cayley-Dickson algebra from the $2^{n}$-dimensional involutive Cayley-Dickson algebra.  A.A. Albert did much to articulate the state of affairs in the early part of the twentieth century, if memory serves.