1
$\begingroup$

I can't imagine a right picture of a linear transformation $\mathbb{F}_{p} \mapsto \mathbb{F}_p$ or $\mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}$ etc (over the field $\mathbb{F}_p$) although they pop up everywhere in algebraic number theory. What's the right mental imagery to imagine about such mappings ?

One way I've tried is to interpret them as lattice model as mentioned in (warning: no LaTeX) https://pdfs.semanticscholar.org/49af/57eafeb2bd17fdf30a7e227f45650fdb46ac.pdf but it's not always helpful.

I tried to use the fact any such map (not necessarily linear) between two finite fields with same characteristic is a polynomial but even then I couldn't successfully use that to visualize stuff.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ linear transformations $\mathbb{F}_{p^k}\to\mathbb{F}_{p^k}$ are $k\times k$ matrices, with entries in $\mathbb{F}_p$, so you can look at their eigenvalue etc. $\endgroup$
    – Dima Pasechnik
    Commented Apr 5, 2019 at 15:10
  • 1
    $\begingroup$ Possibly draw a checkered torus or something like that for the $2 \times 2$-case, and overlay another "skew" lattice on top of it. Not sure how useful visualization generally can be in characteristic $p>0$. We've all seen the $2$-adic solenoid in one or the other artful representation; did any of us learn anything about $2$-adics from that? $\endgroup$
    – darij grinberg
    Commented Apr 5, 2019 at 19:38
  • $\begingroup$ @darijgrinberg Thanks for replying, but then how you in general build intuition about finite fields maps ? The only linear maps I have good intuition is for maps between real spaces with dimension not exceeding three (and the intuition of course coming from just visualizing it). $\endgroup$
    – katana_0
    Commented Apr 5, 2019 at 19:46
  • 1
    $\begingroup$ My intuition rarely ever comes from pictures, so I don't have this particular problem... You can build up some experience with matrices over $\mathbb{F}_2$ by playing lights out (aka "button madness" on old Windowses), and with matrices over $\mathbb{Z} / 26$ (not a field, but close enough) by encrypting and decrypting the Hill cipher; as for basis-free, I don't really have any pictures in my mind. $\endgroup$
    – darij grinberg
    Commented Apr 5, 2019 at 20:09

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .