AI 中文总结
该研究针对32维Cayley-Dickson代数$A_5$，构造14次齐次多项式$D_{14}$并证明左乘法行列式的分解式，得到$A_5$左零因子的判据，还分析了其与十六元数$A_4$对应结构的差异。
AI 中文摘要
我们研究32维Cayley-Dickson代数$A_5$中左乘法的行列式。对$x \in A_5$，记$L_x$为左乘法，$N(x)=|x|^2$。利用$L_{x^*}L_x$的特征空间的四重重数以及应用于其迹不变量的Newton恒等式，我们构造了一个14次齐次多项式$D_{14}$，并证明了分解式$\det L_x=N(x)^2D_{14}(x)^2$。由此可得，$A_5$中的非零元素$x$是左零因子当且仅当$D_{14}(x)=0$。我们还证明了尖锐界$0 \le D_{14}(x) \le N(x)^7$，该界给出$0 \le \det L_x \le |x|^{32}$，并根据交错元素刻画了等号情况。我们将该分解与十六元数$A_4$中的对应结构进行比较：在$A_4$中会出现一个额外的范数因子，而在$A_5$中，我们经复化后证明$N$不整除$D_{14}$，因此从$A_4$到$A_5$的过渡展现了行列式分解的真实变化。该构造给出了$A_5$行列式水平谱结构的与基无关的多项式描述，以及其零因子的显式代数判据。
英文摘要
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.
