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When teaching Algebra, I try to share my fascination about two apparently unrelated questions, which turn out to involve the same theory:

  • classifying the finitely generated abelian groups,
  • classifying the endomorphism of a vector space, up to similarity.

These problems are solved by using the structure of finitely generated modules over a a principal idea domain, $\mathbb Z$ in the first case, $k[X]$ in the second case.

I presume that these problems were first solved independently from each other. But,

who discovered that these are two applications of the same theorem ?

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    $\begingroup$ Maybe a clue could be uncovered in a classic algebra text ( such as Van der Waerden's "Modern Algebra")? Priority is often tricky, but I imagine that whoever first proved the structure of fg modules over a PID in something like its modern form would see those two consequences fairly quickly. $\endgroup$
    – Geoff Robinson
    Commented Dec 5, 2014 at 11:20
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    $\begingroup$ I'm not going to fix the typo "principal idea domain" because I sort of like it as it is. $\endgroup$
    – Andreas Blass
    Commented Dec 5, 2014 at 13:59
  • $\begingroup$ @Andreas. Oups! I did not write that way on purpose :) $\endgroup$
    – Denis Serre
    Commented Dec 5, 2014 at 14:04
  • $\begingroup$ @Geoff. I checked Van der Waerden's book. It does not treat modules. $\endgroup$
    – Denis Serre
    Commented Dec 5, 2014 at 14:05
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    $\begingroup$ The structure theorem goes at least as far back as a 1912 paper of Steinitz in which he proves the structure theorem for f.g. modules over a Dedekind domain. Unfortunately the paper appears to be in German. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 5, 2014 at 18:34

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