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We present a resultant-based method to calculate the overdetermined strata for degree 4 hyperbolic polynomials in one variable. It is a new method to calculate overdetermined strata. We present also the complete study of the overdetermined strata for degree 4 hyperbolic polynomials by the geometric method.

Consider an algebraic number field K of degree n, A K is its ring of integers and a prime number p inert in K. Let F (u 1 , . . . , u n , x) be the generic polynomial of integers of K. We will study in advance the stability of this polynomial and then, we will apply it in order to obtain all the monic irreducible polynomials in F p [x] of degree d dividing n.Mathematics Subject Classification: 11R09, 11T06, 12E10