Dependence of algebraic elements in a finite field Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of $\mathbb{F}_p$). Consider another different monic irreducible polynomial $g(Y)=Y^3+AY^2+BY+C$ in $\mathbb{F}_p[Y]$. Let $y$ be a root of $g(Y)=0$ again in the same closure.
How do we explicity relate $x$ and $y$? Can we express $y$ as a polynomial in $x$ with coefficients from $\mathbb{F}_p$? 
I know that the field extension $\mathbb{F}_{p^3}$ which can be realized as $\mathbb{F}_p[X]/f$ contains $x$ and $y$ (as it contains every root of every irreducible of degree $3$), so $y$ should be expressible in the form 
$$y=\alpha x^2+ \beta x +\gamma$$
for some $\alpha, \beta, \gamma \in \mathbb{F}_p$.
But this is what I am trying to establish, without using the uniqueness of $\mathbb{F}_{p^3}$ or it being the splitting field of $X^{p^3}-X$. I would prefer not using any non-trivial facts about finite fields here. My aim is to show that every cubic irreducible has a root in the specific extension $\mathbb{F}_p[X]/f$, and I want to do this by explicitly obtaining a dependence of the form $y=\alpha x^2+ \beta x +\gamma$ whenever $y$ satisfies $y^3+Ay^2+By+C=0$. 
Is there some algebraic expression or algorithm to find $\alpha, \beta, \gamma$? Something on the lines of resultants maybe.
I was thinking of considering the ideal $<X^3+aX^2+bX+c, Y^3+AY^2+BY+C>$ in $\mathbb{F}_p[X,Y]$ and show that there exists an element in this ideal with total degree at most $2$. But it doesn't seem to lead me anywhere much, especially since I am not clear how to incorporate the (essential) property that both the polynomials are irreducible. Would a Grobner basis of the ideal help with this? I am not too familiar with that technique, but will explore it if it is relevant here. Thanks. 
(This has now been cross-posted to MathOverflow: https://mathoverflow.net/questions/243458/algebraic-dependence-of-irreducibles-in-a-finite-field) 
 A: I am far from an expert in finite fields, and maybe somebody else can give a more satisfactory answer than the very partial attempt I give here. I think that your question brings up a very serious problem, at least for hand computation, and all I can suggest is this:
You are worrying about two irreducible cubics $f$ and $g$ over $k=\Bbb F_p$, and you hope to find any one of the three isomorphisms between $K_1=k[x]/(f$) and $K_2=k[x]/(g)$. (There really is no specific $\Bbb F_{p^3}$, just instantiations like these fields $K_i$.)  Deal with one of the fields, say $K_1$, and now you know that $g$ splits completely over $K_1$. If $\lambda=a_0+a_1\bar x+a_2\bar x^2$ is one of the roots of $g$ in $K_1$, where $\bar x$ is the image of $x$ in $K_1$, you can get the corresponding isomorphism from $K_2$ to $K_1$, by sending $\hat x$ (the image of $x\in K_2$) to $\lambda$. But I think you already knew this. In case $p$ was large, it might be relatively hard (I’m thinking of hand computation here) to find your $\lambda$. There’s nothing unsolvable about the problem here, but I can only suggest very specialized tricks, nothing that would be guaranteed to short-cut the problem.
