Say you're given a ordered set of $n$ relatively prime elements, $a_1,\dots,a_n$ in a principal ideal domain $D$. If I relabel these elements $a_{11},\dots,a_{1n}$ in the same order, is it possible to find some remaining $a_{kj}$ in $D$ such that $(a_{kj})$ is an invertible $n\times n$ matrix over $D$?
Can any set of $n$ relatively prime elements be extended to an invertible matrix?
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linear-algebra
principal-ideal-domains
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0In a more general setting, this became known as Serre's problem, and later the Quillen-Suslin Theorem. – 2011-10-28
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0@Gerry: Can you indicate how this is related to Quillen-Suslin? – 2011-10-29
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0@Soarer, sorry, it's a bit out of my depth. Some statements of the problem/theorem don't look anything like Aria's qiestion, but if you type Quillen Suslin matrix into a search engine you will probably find some references that bring out the relation. – 2011-10-29
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0@Soarer, Here's a reference: Moshe Roitman, Completing unimodular rows to invertible matrices, J. Algebra 49 (1977), no. 1, 206–211, MR0453779 (56 #12033). From the review: Quillen's solution of Serre's problem implies that if $R$ is a commutative ring, then any unimodular row over $R[x]$ which contains a unitary polynomial is completable to an invertible matrix. The author shows.... Other related results on this completion problem are discussed. – 2011-10-31