The [Eilenberg–Niven theorem][1] generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.² 

Does similar theorem holds for the [sedenion algebra][2]?

*References*:

\[1\] <cite authors="Eilenberg, Samuel; Niven, Ivan">_Eilenberg, Samuel; Niven, Ivan_, [**The “fundamental theorem of algebra” for quaternions**](http://dx.doi.org/10.1090/S0002-9904-1944-08125-1), Bull. Am. Math. Soc. 50, 246–248 (1944). [ZBL0063.01228](https://zbmath.org/?q=an:0063.01228).</cite>

\[2\] <cite authors="Jou, Yuh-Lin">_Jou, Yuh-Lin_, The ‘fundamental theorem of algebra’ for Cayley numbers, Sci. Record, Acad. Sinica 3, 29–33 (1950). [ZBL0039.26701](https://zbmath.org/?q=an:0039.26701).</cite>  


  [1]: https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Niven_theorem
  [2]: https://en.wikipedia.org/wiki/Sedenion

Edit: This question is indeed answered in the negative (see the [comment](https://mathoverflow.net/questions/404342/fundamental-theorem-of-algebra-for-sedenions#comment1203293_404342) of user49822). Of course the sedenion algebra has zero divisors, f.i. $a = e_1+e_{10}$. But each $x \in \mathbb{S} \setminus \{0\}$ has an inverse $1/x = \bar{x}/\|x\|^2$ with $x \cdot 1/x = 1/x \cdot x = 1$. In particular $1/a = -a/2$.