Is the unimodular row $(x,y,z)$ completable over the ring $({\mathbb Z}/2{\mathbb Z})[x,y,z,y',z']/\langle x^2+yy'+zz'-1 \rangle$ ?
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1$\begingroup$ A good strategy for getting an answer is defining the terms you are using. What is a unimodular row, what is completable? $\endgroup$– Franz LemmermeyerCommented May 28, 2011 at 13:35
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1$\begingroup$ Also you should post your own ideas / approaches, etc. $\endgroup$– Martin BrandenburgCommented May 28, 2011 at 13:47
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$\begingroup$ In other words, can we find $p_1,p_2,p_3,q_1,q_2,q_3 \in (\mathbb{Z}/2\mathbb{Z})[x,y,z,y',z'] $ such that $$\begin{array}{|ccc|}x &p_1&q_1\\ y&p_2&q_2\\ z &p_3&q_3\\ \end{array} \equiv 1 \; \mod \; \langle x^2+yy'+zz'-1\rangle \; ?$$ $\endgroup$– user15425Commented Jun 3, 2011 at 17:02
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1 Answer
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Your question is equivalent to the $n=2$ case of what T.Y. Lam calls "Murthy's $(a,b,c)$ problem" in his book "Serre's problem on projective modules. (This is statement 5.7 on p. 323 in the 2006 edition).
Lam indicates that this is wide open.
