A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications

Published November 11, 2025 | Version v3

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Euler’s identity $e^{i\theta}=\cos\theta+i\sin\theta$ extends to any element $E$ with

$E^2=-1$ inside a real associative or alternative algebra, but the corresponding

fractional-power formulas, normalization procedures, and geometric behavior remain

scattered across the literature and usually treated on a case-by-case basis.

This paper develops a unified, dimension-independent framework for exponentiation and

fractional powers in real hypercomplex algebras.

Given any element $A$ with scalar square $A^2=\lambda\in\mathbb{R}$, we show that $A$

generates a two-dimensional associative subalgebra.

When $\lambda<0$, normalization yields an imaginary unit

$E=A/\sqrt{-A^2}$ and the closed-form identity

$A^x=(\sqrt{-A^2})^x\Bigl[\cos\Bigl(\tfrac{\pi x}{2}\Bigr)+ E\sin\Bigl(\tfrac{\pi x}{2}\Bigr)\Bigr].$

When $\lambda>0$ (split-algebra case), an analogous hyperbolic formula holds.

This yields a canonical, algebraic definition of $A^x$ for all fractional $x$ without

power-series expansions or analytic continuation.

We give a precise minimal-polynomial criterion for when such closed forms exist, identify

the obstruction locus in higher Cayley--Dickson algebras (non-scalar squares and

zero-divisors), and extend the geometric interpretation of rotations and boosts to

arbitrary alternative algebras.

Finally, we introduce \emph{hypercomplex cyclotomy}: every imaginary direction $E$ with

$E^2=-1$ defines a distinct cyclotomic plane $\mathbb{Q}(\zeta_{n,E})$, with Galois action

realized as geometric rotation.

This produces a continuous family of cyclotomic embeddings not present in the classical

complex theory.

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