
February 24

Nijmegen, DIAMANT intercity: Special day on radical extensions

12:1013:00
 Hendrik Lenstra, Algorithmic Galois theory
[PDF]
Abstract.
Algorithmic Galois theory occupies itself with the design
and analysis of efficient algorithms concerning algebraic
field extensions. The main purpose of the present lecture
is to explain the rules of the game. The best results that
have been obtained are related to solvability by radicals,
and they have been achieved by means of group theory. A
number of open but apparently feasible problems will be
formulated.

13:0014:00

Lunch break

14:0014:50

Bart de Smit, Entangled radicals
[PDF]
Abstract.
For a field K of characteristic 0, a radical group is an
abelian group
B containing K^{*} so that each b in
B has a power in K^{*}.
If all finite subgroups of B are cyclic, then we can embed
B in the multiplicative group of an extension field of
K. To analyze the
radical field extension K(B) of K one needs to
understand
relations between radicals, such as ^{}√5 +
^{}√5 = ^{4}√100.
We will show that these are controlled by the entanglement
group. As an application, we formulate Artin's primitive
root conjecture over number fields.

15:0015:50

Willem Jan Palenstijn,
Computing field degrees of radical extensions
[PDF]
Abstract.
In this talk we will present an algorithm that efficiently computes
the field degree of finite radical extensions over the rationals, up to a
suitably defined cyclotomic part. The main ingredient is the theory
developed in the previous lecture.

15:5016:10

Tea break

16:1017:00

Wieb Bosma,
Some radical algorithms in Magma
Abstract.
We will discuss several problems and examples of
representing certain elements in solvable number fields
as nested radicals in the computer algebra system Magma.
The problems have to do with ambiguities arising
from multivalued root extraction, with testing for equality,
with simplification, and with finding such a representation.



