@article{34843,
  abstract     = {{A polynomial time algorithm to find generators of the lattice of all subfields of a given number field was given in van Hoeij et al. (2013).

This article reports on a massive speedup of this algorithm. This is primary achieved by our new concept of Galois-generating subfields. In general this is a very small set of subfields that determine all other subfields in a group-theoretic way. We compute them by targeted calls to the method from van Hoeij et al. (2013). For an early termination of these calls, we give a list of criteria that imply that further calls will not result in additional subfields.

Finally, we explain how we use subfields to get a good starting group for the computation of Galois groups.}},
  author       = {{Elsenhans, Andreas-Stephan and Klüners, Jürgen}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  pages        = {{1--20}},
  publisher    = {{Elsevier BV}},
  title        = {{{Computing subfields of number fields and applications to Galois group computations}}},
  doi          = {{10.1016/j.jsc.2018.04.013}},
  volume       = {{93}},
  year         = {{2018}},
}

@article{34846,
  abstract     = {{Given a field extension K/k of degree n we are interested in finding the subfields of K containing k. There can be more than polynomially many subfields. We introduce the notion of generating subfields, a set of up to n subfields whose intersections give the rest. We provide an efficient algorithm which uses linear algebra in k or lattice reduction along with factorization in any extension of K. Implementations show that previously difficult cases can now be handled.}},
  author       = {{van Hoeij, Mark and Klüners, Jürgen and Novocin, Andrew}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  pages        = {{17--34}},
  publisher    = {{Elsevier BV}},
  title        = {{{Generating subfields}}},
  doi          = {{10.1016/j.jsc.2012.05.010}},
  volume       = {{52}},
  year         = {{2011}},
}

@article{34900,
  abstract     = {{We describe methods for the computation of Galois groups of univariate polynomials over the rationals which we have implemented up to degree 15. These methods are based on Stauduhar’s algorithm. All computations are done in unramified p -adic extensions. For imprimitive groups we give an improvement using subfields. In the primitive case we use known subgroups of the Galois group together with a combination of Stauduhar’s method and the absolute resolvent method.}},
  author       = {{Geissler, Katharina and Klüners, Jürgen}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  number       = {{6}},
  pages        = {{653--674}},
  publisher    = {{Elsevier BV}},
  title        = {{{Galois Group Computation for Rational Polynomials}}},
  doi          = {{10.1006/jsco.2000.0377}},
  volume       = {{30}},
  year         = {{2000}},
}

@article{34901,
  abstract     = {{Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L / K is cyclic.}},
  author       = {{Acciaro, Vincenzo and Klüners, Jürgen}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  number       = {{3}},
  pages        = {{239--252}},
  publisher    = {{Elsevier BV}},
  title        = {{{Computing Local Artin Maps, and Solvability of Norm Equations}}},
  doi          = {{10.1006/jsco.2000.0361}},
  volume       = {{30}},
  year         = {{2000}},
}

@article{34899,
  abstract     = {{We describe methods for the construction of polynomials with certain types of Galois groups. As an application we deduce that all transitive groups G up to degree 15 occur as Galois groups of regular extensions of ℚ (t), and in each case compute a polynomial f ∈ ℚ [ x ] with Gal(f)  = G.}},
  author       = {{Klüners, Jürgen and Malle, Gunter}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  number       = {{6}},
  pages        = {{675--716}},
  publisher    = {{Elsevier BV}},
  title        = {{{Explicit Galois Realization of Transitive Groups of Degree up to 15}}},
  doi          = {{10.1006/jsco.2000.0378}},
  volume       = {{30}},
  year         = {{2000}},
}

@article{34898,
  abstract     = {{We compute a polynomial with Galois group SL₂(11) over ℚ. Furthermore we prove that SL₂(11) is the Galois group of a regular extension of ℚ (t).}},
  author       = {{Klüners, Jürgen}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  number       = {{6}},
  pages        = {{733--737}},
  publisher    = {{Elsevier BV}},
  title        = {{{A Polynomial with Galois GroupSL2(11)}}},
  doi          = {{10.1006/jsco.2000.0380}},
  volume       = {{30}},
  year         = {{2000}},
}

@article{34902,
  abstract     = {{We present a new polynomial decomposition which generalizes the functional and homogeneous bivariate decomposition of irreducible monic polynomials in one variable over the rationals. With these decompositions it is possible to calculate the roots of an imprimitive polynomial by solving polynomial equations of lower degree.}},
  author       = {{Klüners, Jürgen}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  number       = {{3}},
  pages        = {{261--269}},
  publisher    = {{Elsevier BV}},
  title        = {{{On Polynomial Decompositions}}},
  doi          = {{10.1006/jsco.1998.0252}},
  volume       = {{27}},
  year         = {{1999}},
}

@article{34903,
  abstract     = {{The software packageKANT V4for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.}},
  author       = {{DABERKOW, M. and FIEKER, C. and Klüners, Jürgen and POHST, M. and ROEGNER, K. and SCHÖRNIG, M. and WILDANGER, K.}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  number       = {{3-4}},
  pages        = {{267--283}},
  publisher    = {{Elsevier BV}},
  title        = {{{KANT V4}}},
  doi          = {{10.1006/jsco.1996.0126}},
  volume       = {{24}},
  year         = {{1997}},
}

@article{34904,
  abstract     = {{The purpose of this article is to determine all subfields ℚ(β) of fixed degree of a given algebraic number field ℚ(α). It is convenient to describe each subfield by a pair (h,g) of polynomials in ℚ[t] resp. Z[t] such thatgis the minimal polynomial of β = h(α). The computations are done in unramifiedp-adic extensions and use information concerning subgroups of the Galois group of the normal closure of ℚ(α) obtained from the van der Waerden criterion.}},
  author       = {{Klüners, Jürgen and Pohst, Michael}},
  issn         = {{0747-7171}},
  journal      = {{Journal of Symbolic Computation}},
  keywords     = {{Computational Mathematics, Algebra and Number Theory}},
  number       = {{3-4}},
  pages        = {{385--397}},
  publisher    = {{Elsevier BV}},
  title        = {{{On Computing Subfields}}},
  doi          = {{10.1006/jsco.1996.0140}},
  volume       = {{24}},
  year         = {{1997}},
}