Hurwitz (1898) proved that a real normed algebra must have dimension , 2, 4, or 8. There are four real normed algebras of dimension
2: the complex numbers and three others (Koecher
and Remmert 1988).
Real normed algebras have no zero divisors since the equation
implies that .
Hurwitz, A. "Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln." Nachr. Königl. Gesell. Wiss.
Göttingen. Math.-Phys. Klasse, 309-316, 1898.Koecher, M. and
Remmert, R. "Composition Algebras. HURWITZ's Theorem--Vector-Product Algebras."
Ch. 10 in Ebbinghaus, H.-D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer,
K.; Neukirch, J.; Prestel, A.; and Remmert, R. Numbers.
New York: Springer-Verlag, pp. 265-280, 1988.