Generalization of the Hopf commutative theorem

Abstract

ABSTRACTA famous theorem of Hopf [14] states that: Every finite-dimensional real commutative division algebra A is at most two-dimensional. To this day, no elementary proof of Hopf’s theorem is known. In the present paper, we extend this result to more general situation. Indeed, we show that if A is a real commutative algebraic algebra without divisors of zero, then A is finite dimensional. Moreover, we prove by a simple algebraic method, there is no four-dimensional real commutative division algebra. Finally, we prove that every third-power associative real algebra, with unit element, without divisors of zero, and algebraic of degree ≠8, is quadratic. This last generalizes previously known results of Diankha et al. [6].