ac.commutative algebra - Field structure for R^n

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.

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