Denote by $\mathbb{H}[x\_1,\dots,x\_n]$ the ring of polynomials in $n$ variables with quaternionic coefficients, where the variables **commute** with each other and with the coefficients. Two polynomials $P,Q\in \mathbb{H}[x\_1,\dots,x\_n]$ are *similar*, if $P=a Q b$ for some $a,b\in \mathbb{H}$. A ring $\mathbb{K}$ is *factorial*, if the equality $P\_1\cdot\dots\cdot P\_n=Q\_1\cdot\dots\cdot Q\_m$, where $P\_1,\dots, P\_n,Q\_1,\dots,Q\_m\in \mathbb{K}$ are irreducible (and noninvertible) elements, imply that $n=m$ and there is a permutation $s\in S\_n$ such that $P\_k$ is similar to $Q\_{s(k)}$ for each $k=1,\dots,n$? 


By [1, Theorem 1] and [2, Theorem 2.1] it follows that $\mathbb{H}[x]$ is factorial.



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> Is the ring $\mathbb{H}[x,y]$ factorial?
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This question is a continuation of the following ones:


[When the determinant of a 2x2 polynomial matrix is a square?](https://mathoverflow.net/questions/79063/when-the-determinant-of-a-2x2-polynomial-matrix-is-a-square)


[Pythagorean 5-tuples](https://mathoverflow.net/questions/62820/pythagorean-5-tuples)


[1] Oystein Ore, Theory of non-commutative polynomials, Annals of Math. (II) 34, 1933, 480-508.


[2] Graziano Gentili and Daniele C. Struppa, On the Multiplicity of Zeroes of Polynomials with Quaternionic Coefficients, Milan J. Math. 76 (2008), 15-25, DOI 10.1007/s00032-008-0093-0.



Is the ring of quaternionic polynomials factorial?
Just to remove the question from the 'Unanswered' list:
$(x-i)\cdot((x+i)(y+j)+1)=((y+j)(x+i)+1)\cdot(x-i)$, hence $\mathbb{H}[x,y]$ is not factorial. 


