— Let us consider an algebraic function field defined over a finite Galois extension K of a perfect field k. We recall some elementary conditions allowing the descent of the definition field of the algebraic function field from K to k. We apply these results to the descent of the definition field of a tower of function fields. We give explicitly the equations of the intermediate steps of an Art...

1. The main theorems revisited We now state in more detail the main theorems of class field theory. However, we need two more preliminary definitions: Definition 1.1. Given an extension L/K of number fields, and a fractional ideal I of L, we define the fractional ideal N L/K (I) of K as follows: write I = i q ei i , let p i = q i ∩ O K , and let N L/K (I) = i p eifi i. Definition 1.2. Let σ : K...

Let k be an Hilbertian field, i.e. a field for which Hilbert's irreducibility theorem holds (cf. [1, 5]). It is obvious that the degree of the algebraic closure k of k is infinite with respect to k. It is not obvious that the same is true for the maximal p-extension of k, p a prime number. Let A be a finite abelian group. The question whether there exists a Galoisian extension l/k with Galois g...

Over the last 25 years, a lot of work has been done on seeking for decidable nonregular extensions of Propositional Dynamic Logic (PDL). Only recently, an expressive extension of PDL, allowing visibly pushdown automata (VPAs) as a formalism to describe programs, was introduced and proven to have a satisfiability problem complete for deterministic double exponential time. Lately, the VPA formali...

We study a variant of the inverse problem of Galois theory and Abhyankar’s conjecture. If p is an odd rational prime and G is a finite p-group generated by s elements, s minimal, does there exist a normal extension L/Q such that Gal (L/Q) ∼= G with at most s rational primes that ramify in L? Given a nilpotent group of odd order G with s generators, we obtain a Galois extension L/Q with precisel...

Let S be a closed surface with boundary ∂S and let G be a graph. Let K ⊆ G be a subgraph embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S which coincides on K with the given embedding of K. Minimal obstructions for the existence of embedding extensions are classified in cases when S is the disk or the cylinder. Linear time algorithms are presented that ...

We define an extension L/K of absolutely abelian number fields to be Leopoldt if the ring of integers OL of L is free as a module over the associated order AL/K of L/K. Furthermore, we say that an abelian number field K is Leopoldt if every extension L/K with L/Q abelian is Leopoldt. In this paper, we make progress towards a classification of Leopoldt number fields and extensions. The two main ...

We examine situations, where representations of a finite-dimensional F algebra A defined over a separable extension field K/F , have a unique minimal field of definition. Here the base field F is assumed to be a C1-field. In particular, F could be a finite field or k(t) or k((t)), where k is algebraically closed. We show that a unique minimal field of definition exists if (a) K/F is an algebrai...

We say a tame Galois field extension L/K with Galois group G has trivial Galois module structure if the rings of integers have the property that OL is a free OK [G]-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l so that for each there is a tame Galois field extensio...