##
**On computing subfields.**
*(English)*
Zbl 0886.11072

[This is a joint review for three papers (see also Zbl 0886.11070 and Zbl 0886.11071)]

The first paper (Daberkow et al., Zbl 0886.11070) reads like seven computer salesmen presenting KANT V4. The new idea is to call on algorithms to solve problems of relative subfields without first having to select a subset of the costly computations over \(\mathbb Q\). In 1970, fields of degree \(>4\) were out of reach, while in 1990, fields of degree \(>10\) were still out of reach. Now degree 20 and higher (maybe 1000) seems accessible. The Zassenhaus objectives are again embraced (viz., to study orders, units, class numbers, and Galois groups). The algorithms come from MAGMA through the now traditional sources [H. Cohen, A course in computational algebraic number theory, Grad. Texts Math. 138, Springer (1993; Zbl 0786.11071), and M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encycl. Math. Appl. 30, Cambridge University Press (1989; Zbl 0685.12001)]. The shell KASH is advertised through enticing sample inputs and outputs.

The use of subfield techniques helps in finding Hilbert class fields (by Kummer extensions), solving relative norm problems, and in Galois group techniques for listing subfields. There are some hints of pure mathematical interest, for example the use of approximations to \(L\)-series to find a lower bound on the class number to use with an upper bound obtained by the orders of individual primes (with the help of GRH). The most intriguing item might be the solution of Thue’s equation, which is unexplained together with an arcane reference [Y. Bilu and G. Hanrot, Solving Thue’s equations of high degree, Mathematica Gottingensis, Schriftenreihe des Sonderforschungsbereichs Geometrie und Analysis, Heft 11, Göttingen (1995), (added in 2013) final version in J. Number Theory 60, No. 2, 373–392 (1996; Zbl 0867.11017)].

The second paper (Daberkow, Zbl 0886.11071) presents some of the details of the use of subfields (actually on an earlier version, Kant V2). Norms and units in relative intermediate fields may be used to solve absolute norm problems and find units in fields. When the units of subfields are lifted (to a subgroup of units), large indices may be encountered. Likewise relative integral bases may be projective (not free) for nonprincipal differents. Various cases are illustrated with convincing evidence of better running times with the use of relative subfields.

The third paper (Klüners and Pohst) carries out the determination of all subfields of \(K=\mathbb Q(\alpha)\) (for \(\alpha\) a root of a given polynomial) again using KANT V4. The method is to first determine the Galois group by Van der Waerden’s method (cycles modulo \(p\)) and Hensel lifting. The subfields \(L=\mathbb Q(\beta)\) of \(K\) are listed in terms of \(\beta\) as a polynomial in \(\alpha\) also with the equation for \(\beta\) over \(\mathbb Q\). It is noted that \(K\) is often the Hilbert class field of \(L\). In the examples, the degree of \(K\) runs as high as 24.

The first paper (Daberkow et al., Zbl 0886.11070) reads like seven computer salesmen presenting KANT V4. The new idea is to call on algorithms to solve problems of relative subfields without first having to select a subset of the costly computations over \(\mathbb Q\). In 1970, fields of degree \(>4\) were out of reach, while in 1990, fields of degree \(>10\) were still out of reach. Now degree 20 and higher (maybe 1000) seems accessible. The Zassenhaus objectives are again embraced (viz., to study orders, units, class numbers, and Galois groups). The algorithms come from MAGMA through the now traditional sources [H. Cohen, A course in computational algebraic number theory, Grad. Texts Math. 138, Springer (1993; Zbl 0786.11071), and M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encycl. Math. Appl. 30, Cambridge University Press (1989; Zbl 0685.12001)]. The shell KASH is advertised through enticing sample inputs and outputs.

The use of subfield techniques helps in finding Hilbert class fields (by Kummer extensions), solving relative norm problems, and in Galois group techniques for listing subfields. There are some hints of pure mathematical interest, for example the use of approximations to \(L\)-series to find a lower bound on the class number to use with an upper bound obtained by the orders of individual primes (with the help of GRH). The most intriguing item might be the solution of Thue’s equation, which is unexplained together with an arcane reference [Y. Bilu and G. Hanrot, Solving Thue’s equations of high degree, Mathematica Gottingensis, Schriftenreihe des Sonderforschungsbereichs Geometrie und Analysis, Heft 11, Göttingen (1995), (added in 2013) final version in J. Number Theory 60, No. 2, 373–392 (1996; Zbl 0867.11017)].

The second paper (Daberkow, Zbl 0886.11071) presents some of the details of the use of subfields (actually on an earlier version, Kant V2). Norms and units in relative intermediate fields may be used to solve absolute norm problems and find units in fields. When the units of subfields are lifted (to a subgroup of units), large indices may be encountered. Likewise relative integral bases may be projective (not free) for nonprincipal differents. Various cases are illustrated with convincing evidence of better running times with the use of relative subfields.

The third paper (Klüners and Pohst) carries out the determination of all subfields of \(K=\mathbb Q(\alpha)\) (for \(\alpha\) a root of a given polynomial) again using KANT V4. The method is to first determine the Galois group by Van der Waerden’s method (cycles modulo \(p\)) and Hensel lifting. The subfields \(L=\mathbb Q(\beta)\) of \(K\) are listed in terms of \(\beta\) as a polynomial in \(\alpha\) also with the equation for \(\beta\) over \(\mathbb Q\). It is noted that \(K\) is often the Hilbert class field of \(L\). In the examples, the degree of \(K\) runs as high as 24.

Reviewer: Harvey Cohn (Bowie)

### MSC:

11Y40 | Algebraic number theory computations |

12Y05 | Computational aspects of field theory and polynomials (MSC2010) |

68W30 | Symbolic computation and algebraic computation |

11R32 | Galois theory |

11R37 | Class field theory |