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Algorithms for function fields. (English) Zbl 1116.11325

Summary: Let \(K/\mathbb Q(t)\) be a finite extension. We describe algorithms for computing subfields and automorphisms of \(K/\mathbb Q(t)\). As an application we give an algorithm for finding decompositions of rational functions in \(\mathbb Q(\alpha)\). We also present an algorithm which decides if an extension \(L/\mathbb Q(t)\) is a subfield of \(K\). In case \([K:\mathbb Q(t)] = [L/\mathbb Q(t)]\) we obtain a \(\mathbb Q(t)\)-isomorphism test. Furthermore, we describe an algorithm which computes subfields of the normal closure of \(K/\mathbb Q(t)\).

MSC:

11Y40 Algebraic number theory computations
11R58 Arithmetic theory of algebraic function fields
12E05 Polynomials in general fields (irreducibility, etc.)
12F10 Separable extensions, Galois theory
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References:

[1] Alonso C., J. Symb. Comput. 19 (6) pp 527– (1995) · Zbl 0840.68048
[2] Acciaro V., Math. Comput. 68 pp 1179– (1999) · Zbl 0937.11062
[3] Butler G., Fundamental Algorithms for Permutation Groups. (1991) · Zbl 0785.20001
[4] Daberkow Mario, J. Symb. Comput. 24 (3) pp 267– (1997) · Zbl 0886.11070
[5] Jacobson N., Basic Algebra II. (1980) · Zbl 0441.16001
[6] Kliiners J., Dissertation, in: Über die Berechnung von Automor-phismen und Teilkörpem algebraischer Zahlkörper. (1997)
[7] Kliiners J., Journal de Théorie des Nombres de Bordeaux 10 pp 243– (1998) · Zbl 0935.11047
[8] Klüners Jürgen, J. Symb. Comput. 30 pp 675– (2000) · Zbl 0967.12004
[9] Kliiners J., J. Symb. Comput. 24 (3) pp 385– (1997) · Zbl 0886.11072
[10] Malle Gunter, Inverse Galois Theory. (1999) · Zbl 0940.12001
[11] Pohst Michael E., Math. Comput. 48 pp 177– (1987)
[12] von zur Gathen J., Modem Computer Algebra. (1999) · Zbl 0936.11069
[13] Wielandt H., Finite Permutation Groups. (1964) · Zbl 0138.02501
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