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Details of talk

TitleComputation of Geometric Galois Groups and Absolute Factorizations
PresenterNicole Sutherland (University of Sydney)
Author(s)Nicole Sutherland
SessionPosters
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.