Abstract
The aim of this chapter is an introduction to elementary algorithms in algebraic extensions, mainly over ℚ and, to some extent, over GF(p). We will talk about arithmetic in ℚ(α) and GF(p n) in Section 1 and some polynomial algorithms with coefficients from these domains in Section 2. Then, we will consider the field K of all algebraic numbers over ℚ and show constructively that K indeed is a field, that multiple extensions can be replaced by single ones and that K is algebraically closed, i.e. that zeros of algebraic number polynomials will be elements of K (Section 4–6). For this purpose we develop a simple resultant calculus which reduces all operations on algebraic numbers to polynomial arithmetic on long integers together with some auxiliary arithmetic on rational intervals (Section 3). Finally, we present some auxiliary algebraic number algorithms used in other chapters of this volume (Section 7). This chapter does not include any special algorithms of algebraic number theory. For an introduction and survey with an extensive bibliography the reader is referred to Zimmer [15].
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References
Arnon, D. S.: Algorithms for the Geometry of Semi-Algebraic Sets., Ph.D. Thesis, Univ. of Wisconsin, Madison, 1981.
Calmet, J.: A SAC-2 Implementation of Arithmetic and Root Finding over Large Finite Fields. Interner Bericht, Universität Karlsruhe 1981 (forthcoming).
Calmet, J., Loos, R.: An Improvement of Robin’s Probabilistic Algorithm for Generating Irreducible Polynomials over GF(p). Information Processing Letters 11, 94–95 (1980).
Cauchy, A.: Analyse Algébrique. Oeuvres completes, II serie, tome III, p. 398, formula (48). Paris: 1847.
Collins, G. E.: The Calculation of Multivariate Polynomial Resultants. J. ACM 18, 515–532 (1971).
Collins, G. E., Horowitz, E.: The Minimum Root Separation of a Polynomial. Math. Comp. 28, 589–597 (1974).
Heindel, G. E.: Integer Arithmetic Algorithms for Polynomial Real Zero Determination. J. ACM 18, 533–548 (1971).
Householder, A. S.: Bigradients and the Problem of Routh and Hurwitz. SIAM Review 10, 57–78 (1968).
Householder, A. S., Stuart III, G. W.: Bigradients, Hankel Determinants, and the Padé Table. Constructive Aspects of the Fundamental Theorem of Algebra (Dejon, B., Henrici, P., eds.). London: 1963.
Lenstra, A. K.: Factorisatie van Polynomen. Studieweek Getaltheorie en Computers, Mathematisch Centrum, Amsterdam, 95–134 (1980).
Loos, R.: Computing Rational Zeros of Integral Polynomials by p-adic Expansion. Universität Karlsruhe (1981) (to appear in SIAM J. Comp.).
Pinkert, J. R.: Algebraic Algorithms for Computing the Complex Zeros of Gaussian Polynomials. Ph.D. Thesis, Comp. Sci. Dept., Univ. of Wisconsin, May 1973.
Rabin, M. O.: Probabilistic Algorithms in Finite Fields. SIAM J. Comp. 9, 273–280 (1980).
van der Waerden, B. L.: Algebra I. Berlin-Heidelberg-New York: Springer 1971.
Zimmer, H.: Computational Problems, Methods, and Results in Algebraic Number Theory. Lecture Notes in Mathematics, Vol. 262. Berlin-Heidelberg-New York: Springer 1972.
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© 1982 Springer-Verlag Wien
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Loos, R. (1982). Computing in Algebraic Extensions. In: Buchberger, B., Collins, G.E., Loos, R. (eds) Computer Algebra. Computing Supplementum, vol 4. Springer, Vienna. https://doi.org/10.1007/978-3-7091-3406-1_12
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DOI: https://doi.org/10.1007/978-3-7091-3406-1_12
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-81684-4
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