Intercity Number Theory Seminar
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February 24 Nijmegen, DIAMANT intercity: Special day on radical extensions
12:10-13:00 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.
13:00-14:00 Lunch break
14:00-14:50 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.
15:00-15:50 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.
15:50-16:10 Tea break
16:10-17:00 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.