11

I'm now a TA on an undergraduate course "Algebra II" and the main topics of the course are "rings and modules" and "fields and the Galois theory". We shall cover the fundamental result, namely the structure theorem of finitely generated module over PIDs.

Two typical applications of the theorem are

  • The structure theorem of finitely generated abelian groups (as $\mathbb{Z}$-modules);
  • The canonical forms (including Smith canonical forms and Jordan canonical forms) in linear algebra (by viewing the finite dimensional $k$-vector space $V$ as a $k[\lambda]$-module).

In this post, I'm hoping to collect some interesting applications of the structure theorem besides the two examples above, since the two examples can be seen in almost all textbooks, and these may be boring for students.

For example, this is an interesting application: Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID. Although this is somehow a special case of the "canonical form" example, this is rather surprising and can be regarded as a good example for me.

And moreover, although I'm actually collecting examples for the course, any applications beyond the scope of the course are still great! For example, as @KCd hinted in the comment, the theorem can be seen in basic algebraic number theory all the time, such as the Dirichlet unit theorem.

Thank you all in advance! And thanks @KCd for his helpful comments!

Hetong Xu
  • 2,107
  • 1
    The classification of solutions to homogeneous linear (constant coefficient) recurrence relations is pretty similar to the differential equations example, and the two can be related using generating functions. E.g. you can write down the closed form of the Fibonacci numbers. – Qiaochu Yuan Oct 23 '20 at 01:52
  • 3
    This is used all the time in basic algebraic number theory, but that is beyond the scope of your course and students would not know the terminology. Keep in mind that not everyone will consider canonical forms to be “interesting” applications. When I was a student I did not find those canonical forms to be compelling applications (it all seemed kind of tedious at the time). Also, often what is important is not the classification of f.g. modules on their own, but how they can be “aligned” with a submodule. – KCd Oct 23 '20 at 07:42
  • @KCd Thank you for your comment! Actually, I'm hoping to gain more examples on the application of the structure theorem, which is more surprising or inspiring. In almost all textbooks, authors often use the two examples above as applications, so maybe students are rather bored on them (at least for me). It is the same to me that I did not find those canonical forms rather "interesting" when I learnt it one year ago. Sincerely sorry for not making my ideas clear in the post. Also, could you explain more on your last sentence:" ... but how they can be 'aligned' with a submodule"? – Hetong Xu Oct 23 '20 at 08:19
  • @QiaochuYuan Thank you for your examples! – Hetong Xu Oct 23 '20 at 08:21
  • By “aligned” I mean that if $M$ is finite free over the PID $A$ and $M’$ is a submodule of $M$ then there is a basis $e_1,\ldots,e_n$ of $M$ and nonzero $a_1,\ldots,a_k$ in $A$ such that $a_1e_1,\ldots,a_ke_k$ is a basis of $M’$. – KCd Oct 23 '20 at 12:41
  • @KCd It seems that this result is covered in one of your great expository papers, i.e. Theorem 2.10 in https://kconrad.math.uconn.edu/blurbs/linmultialg/modulesoverPID.pdf . Thank you so much for your comments! And moreover, your expository papers are always of great help in various topics!! I have had tons of aha moments when reading these, which are with carefully explained details and always with inspiring examples! – Hetong Xu Oct 23 '20 at 14:43

0 Answers0