Friday, 9 June 2017

abstract algebra - Dependence of algebraic elements in a finite field

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,γFp.



But this is what I am trying to establish, without using the uniqueness of Fp3 or it being the splitting field of Xp3X. 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 $in\mathbb{F}_p[X,Y]andshowthatthereexistsanelementinthisidealwithtotaldegreeatmost2$. 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)

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