Splitting of quartic polynomials

Abstract:

For integers r, s, t, u define the recursion $A(n + 4) = rA(n + 3) - sA(n + 2) + tA(n + 1) - uA(n)$ where the initial conditions are set up in such a way that $A(n) = {\alpha ^n} + {\beta ^n} + {\gamma ^n} + {\delta ^n}$ where $\alpha ,\beta ,\gamma ,\delta$ are the roots of the associated polynomial $f(x) = {x^4} - r{x^3} + s{x^2} - tx + u$ In this paper a detailed deterministic procedure using the $A(n)$ for finding how $f(x)$ splits modulo a prime integer p is given. This gives for p not dividing the discriminant of $f(x)$ the splitting of p in the field obtained by adjoining a root of $f(x)$ to the rational numbers. There is an interesting connection between the results here for reciprocal polynomials and some work of D. Shanks.