Determinant Factorization of Left Multiplication in the 32 Dimensional Cayley Dickson Algebra

Abstract

We study the determinant of left multiplication in the 32-dimensional Cayley-Dickson algebra $A_5$. For $x \in A_5$, let $L_x$ denote left multiplication and let $N(x)=|x|^2$. Using the fourfold multiplicity of the eigenspaces of $L_{x^*}L_x$ and Newton identities applied to its trace invariants, we construct a homogeneous polynomial $D_{14}$ of degree 14 and prove the factorization $\det L_x=N(x)^2D_{14}(x)^2$. Consequently, a nonzero element $x \in A_5$ is a left zero divisor if and only if $D_{14}(x)=0$. We also prove the sharp bound $0 \le D_{14}(x) \le N(x)^7$, which yields $0 \le \det L_x \le |x|^{32}$, and characterize the equality case in terms of alternative elements. We compare this factorization with the corresponding structure in the sedenions $A_4$. In $A_4$ an additional norm factor occurs, whereas in $A_5$ we prove, after complexification, that $N$ does not divide $D_{14}$. Thus the passage from $A_4$ to $A_5$ exhibits a genuine change in the determinant factorization. The construction gives a basis-independent polynomial description of the determinant-level spectral structure of $A_5$ and an explicit algebraic criterion for its zero divisors.