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Since all the finite field of $p^n$ elements are the splitting field of the separable polynomial $x^{p^n}-x$, all of them are isomphic. In particular if $f_1(x),f_2(x)$ are irreducible polynomials over $\mathbb{F}_p[x]$ of the same degree. Then: $$ \mathbb{F}_p[x]/(f_1(x)) \cong \mathbb{F}_p[x]/(f_2(x))$$ But I want to find an explicit isomorphism. I don't know if it's always possible. But the following could be useful.

Let's consider $f_2$ as a polynomial in $(\mathbb{F}_p[x]/(f_1(x)))[y]$. If $\gamma(x) $ is a root of $f_2(y)$ (a root on $\mathbb{F}_p[x]/(f_1(x))$). Then the following map is an isomorphism: $$\mathbb{F}_p[x]/(f_2(x)): \to \mathbb{F}_p[x]/(f_1(x)) $$ $$x\to \gamma(x)$$

My question if there are techniques to find that $\gamma(x)$.

For example if $f_1 = x^4+x^3+1 , f_2 = x^4+x+1 $ are over $\mathbb{F}_2$ then $\gamma(x)=x^3+x^2 $ it's a root.

If the solution of the general case it's not possible (or unsolved) or too difficult, I want to know at least this particular case :/ I want to compute it in the case $ f_1= x^2+2x+2 , f_2 = x^2+x+3 $ over $\mathbb{F}_7$

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3 Answers 3

up vote 1 down vote accepted

In the field ${\mathbb F}_7[w]$ where $w^2 + 2 w + 2 = 0$, you want to find $\alpha$ such that $\alpha^2 + \alpha + 3 = 0$. It must be of the form $a + b w$ with $a, b \in {\mathbb F}_7$. Well, $(a+b w)^2 + a + b w + 3 = {a}^{2}+3+a-2\,{b}^{2}+ \left( b+2\,ab-2\,{b}^{2} \right) w$, so we want $ {a}^{2}+3+a-2\,{b}^{2} = 0$ and $ b+2\,ab-2\,{b}^{2} = 0$ mod 7. The solutions are $a=2,b=6$ and $a=4,b=1$.

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Are you sure that this result it's correct? Maybe I'm doing something wrong but, here's my understanding of the problem: I have to prove that for example $ \gamma(x)=2x+6 $ it's a root , of the polynomial $f_2$ over the field $ \mathbb{F}/(f_1(x)) $ Computing $f_2(\gamma ) = (2x+6)^2 + (2x+6) + 3 = 4x^2 + 36 + 24x + 2x+ 9 = 4x^2+ 5x + 3 $ But this is not cero $mod (f_1(x)) $ ( since it's clearly not a multiply of the polynomial $f_1$ ) . –  Daniel Sep 27 '12 at 14:31
    
But it does not matter, but thanks to your idea I did the problem , Thanks! –  Daniel Sep 27 '12 at 14:43
    
It's not $2x+6$, it's $2+6x$. $(2+6x)^2+(2+6x)+3 = 36 x^2 + 30 x + 9 \equiv 36 (x^2 + 2 x + 2) \mod 7$. –  Robert Israel Sep 27 '12 at 17:43

You can fctorize $f_2(y)\in\mathbb{F}_p[x]/(f_1(x))$ into linear facrors using factorization algorithms like Cantor–Zassenhaus algorithm or Berlekamp's algorithm . I think there are no easier way to do this in the general case.$ {} {} {} $

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At least for the case $n=2$, I believe the following works.

Let $q(x)$ and $r(x)$ be your irreducible polynomials. Without loss of generality, they're monic. We can complete the square and write them as $q(x) = (x+a)^2 + b$ and $r(x) = (x+c)^2 + d$.

Choose an automorphism of $\mathbb{F}_{p}$ called $\phi$ such that $\phi(b) = d$ (namely, multiplication of elements in $\mathbb{F}_p$ by $db^{-1}$). This automorphism induces an automorphism $\phi^\star$ of $\mathbb{F}_{p} [x]$ given by $$\phi^\star (a_n x^n + \cdots + a_0) = \phi(a_n) x^n + \cdots + \phi(a_0)$$

Let $k = c- \phi(a)$.

Then, the map $a + (q(x)) \mapsto \phi^\star (a) (x+k) + (r(x))$ is an isomorphism between $\mathbb{F}_{p}[x]/((q(x))$ and $\mathbb{F}_{p}[x]/((r(x))$.

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