Talk:Conway polynomial (finite fields)

Norm-compatibility condition

I think the norm-compatibility condition stated as $f_{m}(x)$ divides $f_{n}(x^{r})$ is wrong. In my opinion it has to be $f_{n}(x)$ divides $f_{m}(x^{r})$. This is also the form stated on Frank Lübeck's homepage. Can anybody confirm this ? If so, we can correct it. Hart15 (talk) 17:58, 14 June 2020 (UTC)

I agree that it should be changed. If $\alpha ^{r}$ is a root of $f_{m}(x)$ then $\alpha$ is a root of $f_{m}(x^{r})$. Since $f_{n}(x)$ is the minimal polynomial for $\alpha$, it must divide $f_{m}(x^{r})$. As a sanity check, try the examples $f_{2,2}(x)=x^{2}+x+1$ and $f_{2,4}(x)=x^{4}+x+1$. For these polynomials, $r=(2^{4}-1)/(2^{2}-1)=5$ and one can check that $x^{4}+x+1$ divides $f_{2,2}(x^{5})=x^{10}+x^{5}+1$, while $x^{2}+x+1$ does not divide $f_{2,4}(x^{5})=x^{20}+x^{5}+1$.

It looks like this error was fixed once before, but the error got reintroduced in a later edit. Will Orrick (talk) 22:54, 14 June 2020 (UTC)

I somehow neglected to come back and correct this back in 2020. I have now done so. Will Orrick (talk) 02:53, 30 November 2023 (UTC)

Examples requested

An editor has asked that examples be added: "...the total lack of explicit examples of Conway polynomial makes the article totally unundestandable to people who never had to manipulate them...". For anyone who wants to help, explicit polynomials may be found at the bottom of Frank Luebeck's page, which is linked in the article. What's needed, of course, is not the polynomials by themselves, but an explanation of how they satisfy the definition. This is not something I feel capable of providing at the moment, so I hope someone else will give it a go.

This is a technical topic in the purest sense of the word: the polynomials' main use seems to be in computation. They don't appear to have much conceptual appeal. Perhaps the article is simply too technical and too obscure for Wikipedia, as there don't appear to be qualified editors available to maintain it. If it really is incomprehensible, it could simply be deleted. There is a prehistory of this page, which I can no longer recall how to access. The version that existed prior to what's listed on the history page suffered from copyright violation issues, if I recall correctly. I believe that version was essentially a cut-and-paste of Frank Luebeck's page, which it was claimed was done with Luebeck's permission. Will Orrick (talk) 17:34, 29 November 2023 (UTC)

OK, so there are at least some examples now. If I remain motivated, I may add more. As I alluded to above, I wrote this article because its predecessor was threatened with extinction and I wanted to preserve the topic on Wikipedia, not because I have adequate knowledge of the topic. I hope that some of the Wikipedia regulars who are experts on finite fields will at least take a look at what's here, and make improvements or additions as appropriate. Will Orrick (talk) 17:01, 30 November 2023 (UTC)

I came to this article because it was linked to in Finite field. As the link was motivated with very vague reasons (coherency in some explicit computation), I have replaced the motivation by an explicit description of what Conway polynomial are for. This could be used here for explaining the context of this article. D.Lazard (talk) 19:16, 2 December 2023 (UTC)