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Model polynomials of $GF(p)$ as polynomials of $GF(p^n)$

asked 2023-02-02 14:42:06 +0200

Mairon gravatar image

updated 2023-02-06 19:25:35 +0200

dan_fulea gravatar image

Imagine that we have polynomials $$F_1, F_2, \ldots, F_n : \mathbb{F}_p \rightarrow \mathbb{F}_p\ ,$$ $n$ polynomials in $n$ variables, which define a map of sets $\mathbb{F}_q \rightarrow \mathbb{F}_q$, $q = p^n$, and we want to find the map as a single univariate polynomial over $\mathbb{F}_q$. This is a follow-up question that is answered already here for the reverse direction.

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The reverse direction of the previous question would be taking $n$ polynomials in $n$ variables $F_1, F_2, \ldots, F_n : \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ which (given a choice of generator for $\mathbb{F}_{p^n}$) define a map of sets $\mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$, and trying to realize that map as a single univariate polynomial over $\mathbb{F}_{p^n}$.

rburing gravatar imagerburing ( 2023-02-02 18:32:09 +0200 )edit

Yes, I will update the question with the better description. But that's what I had in mind, is it possible?

Mairon gravatar imageMairon ( 2023-02-03 10:34:46 +0200 )edit

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answered 2023-02-02 15:43:34 +0200

Max Alekseyev gravatar image

Just use .change_ring() method:

p = 7
R.<x> = GF(p)[]
f = x^2 + x + 1
g = f.change_ring(GF(p^3))
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Thank you, I didn't explain the question clearly, I update it.

Mairon gravatar imageMairon ( 2023-02-03 10:39:09 +0200 )edit

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Asked: 2023-02-02 14:42:06 +0200

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Last updated: Feb 06 '23