Lets work over the finite field Fp for a prime p. Consider a monic irreducible polynomial f(X)=X3+aX2+bX+c in Fp[X]. Let x be a root of f(x)=0 (say, in the closure of Fp). Consider another different monic irreducible polynomial g(Y)=Y3+AY2+BY+C in Fp[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 Fp?
I know that the field extension Fp3 which can be realized as Fp[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=αx2+βx+γ
for some α,β,γ∈Fp.
But this is what I am trying to establish, without using the uniqueness of Fp3 or it being the splitting field of Xp3−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 Fp[X]/f, and I want to do this by explicitly obtaining a dependence of the form y=αx2+βx+γ whenever y satisfies y3+Ay2+By+C=0.
Is there some algebraic expression or algorithm to find α,β,γ? Something on the lines of resultants maybe.
I was thinking of considering the ideal $
(This has now been cross-posted to MathOverflow: https://mathoverflow.net/questions/243458/algebraic-dependence-of-irreducibles-in-a-finite-field)
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