Splitting of quartic polynomials

Author:
William W. Adams

Journal:
Math. Comp. **43** (1984), 329-343

MSC:
Primary 12E10; Secondary 11R09, 11R27

DOI:
https://doi.org/10.1090/S0025-5718-1984-0744941-3

MathSciNet review:
744941

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Abstract: For integers *r, s, t, u* define the recursion where the initial conditions are set up in such a way that where are the roots of the associated polynomial In this paper a detailed deterministic procedure using the for finding how splits modulo a prime integer *p* is given. This gives for *p* not dividing the discriminant of the splitting of *p* in the field obtained by adjoining a root of to the rational numbers. There is an interesting connection between the results here for reciprocal polynomials and some work of D. Shanks.

**[1]**W. W. Adams & D. Shanks, "Strong primality tests that are not sufficient,"*Math. Comp.*, v. 39, 1982, pp. 255-300. MR**658231 (84c:10007)****[2]**E. Berlekamp, "Factoring polynomials over finite fields,"*Bell System Tech. J.*, v. 46, 1967, pp. 1853-1859. MR**0219231 (36:2314)****[3]**L. Carlitz, "A special quartic congruence,"*Math. Scand.*, v. 4, 1956, pp. 243-246. MR**0090601 (19:837e)****[4]**B. N. Delone & D. K. Fadeev,*The Theory of Irrationalities of the Third Degree*, Transl. Math. Monographs, vol. 10, Amer. Math. Soc., Providence, R. I., 1964. MR**0160744 (28:3955)****[5]**D. E. Knuth,*Seminumerical Algorithms*, 2nd ed., Addison-Wesley, Reading, Mass., 1980. MR**633878 (83i:68003)****[6]**S. Schwarz, "Sur le nombre des racines et des facteurs irréductibles d'une congruence donnée,"*Časopis Pěst. Mat. Fys.*, v. 69, 1940, pp. 128-145. MR**0004818 (3:66a)****[7]**D. Shanks, "Dihedral quartic approximations and series for ,"*J. Number Theory*, v. 14, 1982, pp. 397-423. MR**660385 (83k:12010)****[8]**D. Shanks,*Prime-Splitting in Associated Cubic and Quartic Fields*:*Some Implications and Some Techniques*. (To appear.)

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DOI:
https://doi.org/10.1090/S0025-5718-1984-0744941-3

Article copyright:
© Copyright 1984
American Mathematical Society