Is the ring of quaternionic polynomials factorial? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:35:59Z http://mathoverflow.net/feeds/question/89094 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89094/is-the-ring-of-quaternionic-polynomials-factorial Is the ring of quaternionic polynomials factorial? mikhail skopenkov 2012-02-21T10:44:05Z 2012-03-05T14:07:56Z <p>Denote by $\mathbb{H}[x_1,\dots,x_n]$ the ring of polynomials in $n$ variables with quaternionic coefficients, where the variables <strong>commute</strong> with each other and with the coefficients. Two polynomials $P,Q\in \mathbb{H}[x_1,\dots,x_n]$ are <em>similar</em>, if $P=a Q b$ for some $a,b\in \mathbb{H}$. A ring $\mathbb{K}$ is <em>factorial</em>, 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 $k=1,\dots,n$? </p> <p>By [1, Theorem 1] and [2, Theorem 2.1] it follows that $\mathbb{H}[x]$ is factorial.</p> <blockquote> <p>Is the ring $\mathbb{H}[x,y]$ factorial?</p> </blockquote> <p>This question is a continuation of the following ones:</p> <p><a href="http://mathoverflow.net/questions/79063/when-the-determinant-of-a-2x2-polynomial-matrix-is-a-square" rel="nofollow">http://mathoverflow.net/questions/79063/when-the-determinant-of-a-2x2-polynomial-matrix-is-a-square</a></p> <p><a href="http://mathoverflow.net/questions/62820/pythagorean-5-tuples" rel="nofollow">http://mathoverflow.net/questions/62820/pythagorean-5-tuples</a></p> <p>[1] Oystein Ore, Theory of non-commutative polynomials, Annals of Math. (II) 34, 1933, 480-508.</p> <p>[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.</p> http://mathoverflow.net/questions/89094/is-the-ring-of-quaternionic-polynomials-factorial/90281#90281 Answer by mikhail skopenkov for Is the ring of quaternionic polynomials factorial? mikhail skopenkov 2012-03-05T14:07:56Z 2012-03-05T14:07:56Z <p>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. </p>