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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Splitting of quartic polynomials

Author: William W. Adams
Journal: Math. Comp. 43 (1984), 329-343
MSC: Primary 12E10; Secondary 11R09, 11R27
MathSciNet review: 744941
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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.

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Article copyright: © Copyright 1984 American Mathematical Society

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