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The multiplicative group of $\mathbb{F}_{p^n}$ is the circle group $C _{p^n-1}$. So, each nonzero element corresponds to a rotation, and multiplication composes rotations.

The field is also an extension field of $\mathbb{F}_{p}$, which is equivalent to the field of integers mod $p$ with normal addition and multiplication. Here, the additive group is $C _{p}$ and the multiplicative group is $C _{p-1}$.

The field extension is a vector space over $\mathbb{F}_{p}$, so each addition is the same as pointwise addition over $C _{p}^n$, and explains why repeated addition will always take equal elements to $0$, however the field structure somehow "joins" these together into one larger circle group under multiplication.

How does the field structure link $C _{p}^n$ and $C _{p^n}$? Clearly $(0, 0, ..., 0)$ are joined together as the field additive identity, and multiplication is an interaction between these groups, but I can't see much more than that. I'm especially interested in fields of order $2^n$, even more so $2^{2^n}$. I wouldn't be surprised if different orders lead to very different structures.

chbaker0
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  • I'm not sure what kind of answer you're looking for. Even in the simple situation $n=1$, so that we're looking at addition and multiplication in $\Bbb F_p$, it is simultaneously true that the multiplication is "normal" but also "composes rotations", since the multiplicative group of $\Bbb F_p$ is also cyclic. What would you consider a satisfying answer to "how do addition and multiplication in $\Bbb F_p$ interact"? – Greg Martin Sep 05 '23 at 05:39
  • Apart from the obvious embedding $C_{p^n-1}\to\operatorname{Aut}(C_p^n)$? – user10354138 Sep 05 '23 at 05:40
  • @GregMartin I’m asking about $\mathbb{F}_{p^n}$. Of course it’s obvious with prime order. It’s not obvious to me what addition “means” in prime power fields, when thinking of mult as rotation. And when looking at addition as vector addition, I don’t know what multiplication would mean. These might be obvious with a strong math background but I don’t have that – chbaker0 Sep 05 '23 at 06:23
  • To be clear, “there is no consistent interpretation” is a perfectly fine answer to me. I know discrete math is weird – chbaker0 Sep 05 '23 at 06:24
  • @user10354138 ok, this makes sense (but wasn’t obvious to me), though I have to think more on how to reconcile the two. It’s hard for me to see how to represent the field elements in a way that reflects both. The usual polynomial representation doesn’t make that clear (to me) – chbaker0 Sep 05 '23 at 06:43
  • I advise you to ask in the search window of Stack exchange "vizualisation finite fields" : you will get a lot of interesting things like this text (but maybe, it is not this kind of stuff you are looking for). – Jean Marie Sep 05 '23 at 08:56
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    I guess to me the interaction between addition and cyclic-structure multiplication is not obvious with prime order (it's tied to the discrete logarithm problem for one thing). So if you think that's obvious but the prime-power-field case isn't, I don't understand what shape of answer you want with your question. – Greg Martin Sep 05 '23 at 15:41
  • I have always found Nimber arithmetic to be a nice (but mostly useless) description of the arithmetic of the fields $\Bbb{F}_{2^{2^n}}$. Use the definitions in WP (due to Conway) to build addition and multiplication tables for the set of natural numbers $0,1,\ldots,2^{2^n}-1$. The surprise is that you get a field! Looking at the definition alone it is not immediately obvious that addition = bitwise XOR, but that's the NIM magic. – Jyrki Lahtonen Sep 05 '23 at 18:16
  • One thing caught my eye in your question. There is nothing like $C_{p^n}$ in $\Bbb{F}{p^n}$. Only the additive group $\simeq C_p^n$ and the multiplicative group $\simeq C{p^n-1}$. To check: you do know that $x+x=0$ for every $x\in\Bbb{F}_{2^n}$ and all $n$? – Jyrki Lahtonen Sep 05 '23 at 18:18
  • @JyrkiLahtonen sorry, that was a typo, I meant to put $p^n - 1$ there. I’ll think more about these replies. – chbaker0 Sep 05 '23 at 23:16
  • @Jirki Lahtonen May I ask you which book you are refering to by the initials WP ? – Jean Marie Sep 06 '23 at 06:15
  • @JeanMarie I think he just mean wikipedia, but the relevant Conway book is "Winning ways for your mathematical plays" – chbaker0 Sep 06 '23 at 08:12
  • @chbaker0 Thank you very much. – Jean Marie Sep 06 '23 at 08:17

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