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Details of talk

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

An algorithm being developed to compute geometric Galois groups of polynomials
$\mathbb{Q}(t)$ 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


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

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
calculations over $\mathbb{C}$.
This approach is different to those described in  Ch{\`e}ze and Galligo and
other publications.