# Details of talk

Title | Computation of Geometric Galois Groups and Absolute Factorizations |
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Presenter | Nicole Sutherland (The University of Sydney) |

Author(s) | Nicole Sutherland |

Session | Posters |

Time | |

Abstract | An algorithm being developed to compute geometric Galois groups of polynomials over $\mathbb{Q}(t)$ is discussed. This algorithm also computes a field extension such that the factorization of a polynomial over this field extension is the absolute factorization of the polynomial. Building on the algorithm to compute Galois groups of polynomials over $\mathbb{Q}$ by Fieker and Kl\"uners, the algorithm to compute Galois groups of polynomials over $\mathbb{Q}(t)$ described in Krumm and Sutherland and the algorithm to compute Galois groups of reducible polynomials described in my previous publications, I present an algorithm to compute the geometric Galois group of a polynomial over $\mathbb{Q}(t)$. \def\Gal{\mathop{\mathrm{Gal}}\nolimits} I compute over the algebraic closure $K$ of $\mathbb{Q}$ in the splitting field $\Gamma$ of $f$. Since $\Gal(f/K(t)) = \Gal(f/\mathbb{C}(t))$, I compute the geometric Galois group of $f$ as the Galois group of $f$ considered as a polynomial over $K(t)$. The task is to determine which subgroup of $\Gal(f)$ fixes $K(t)$. I am aware of some results computed by hand by J\"urgen Kl\"uners. However, in all of the cases I have access to these geometric Galois groups are the same as the Galois group over $\mathbb{Q}(t)$. I have begun to implement the algorithm I describe using the computational algerba system {\sc Magma}, and will illustrate my progress with an example in which the geometric Galois group is a non-trivial proper subgroup of the Galois group. I will also mention how geometric Galois groups are linked to inverse Galois theory. Once I compute the algebraic closure $K$ in the computation of the geometric Galois group I can factor polynomials over this field using existing functionality of {\sc Magma}. Thus I can compute the absolute factorization of $f\in\mathbb{Q}(t)[x]$ without using calculations over $\mathbb{C}$. This approach is different to those described in Ch{\`e}ze and Galligo and other publications. |