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In this paper we study couples of finite separable extensions of the function field F q (T) which are arithmetically equivalent, i.e. such that prime ideals of F q [ T ] decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of F q (T) , which are fields such that their field of constants may be bigger than F q. In the second part, we explicitly produce examples of non-geometric extensions of F 2 (T) which are equivalent and non-isomorphic over F 2 (T) and non-equivalent over F 4 (T) , solving a particular Inverse Galois Problem. [ABSTRACT FROM AUTHOR]

This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite set of arbitrary size may ramify. We then continue by examining a conjecture of Harbater that the minimal number of generators of the Galois group of a tame, Galois extension of the rational numbers is bounded above by the sum of a constant and the logarithm of the product of the ramified primes. We prove the validity of Harbater's conjecture in a number of cases, including the situation where we restrict our attention to finite groups containing a nilpotent subgroup of index 1 , 2 , or 3. We also derive some consequences that are implied by the truth of this conjecture. [ABSTRACT FROM AUTHOR]

In this paper we prove that for each integer of the form n = 4 ϖ (where ϖ is a prime between 17 and 73) at least one of the following groups: P Ω n ± (F ℓ s ) , PSO n ± (F ℓ s ) , PO n ± (F ℓ s ) or PGO n ± (F ℓ s ) is a Galois group of Q for almost all primes ℓ and infinitely many integers s > 0. This is achieved by making use of the classification of small degree representations of finite simple groups of Lie type in defining characteristic due to Lübeck and a previous result of the author on the image of the Galois representations attached to RAESDC automorphic representations of GL n (A Q). [ABSTRACT FROM AUTHOR]

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of ℓ -adic Galois representations valued in a semisimple group G using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result of Larsen on Galois images for compatible systems. [ABSTRACT FROM AUTHOR]

In previous works [18] and [19] , we described algorithms to compute the number field cut out by the mod ℓ representation attached to a modular form of level N = 1 . In this article, we explain how these algorithms can be generalised to forms of higher level N . As an application, we compute the Galois representations attached to a few forms which are supersingular or admit a companion mod ℓ with ℓ = 13 and ℓ = 41 , and we obtain previously unknown number fields of degree ℓ + 1 whose Galois closure has Galois group PGL 2 ( F ℓ ) and a root discriminant that is so small that it beats records for such number fields. Finally, we give a formula to predict the discriminant of the fields obtained by this method, and we use it to find other interesting examples, which are unfortunately out of our computational reach. [ABSTRACT FROM AUTHOR]

We prove the existence of certain rationally rigid triples in F 4 ( p ) for good primes p (i.e., p > 3 ), thereby showing that these groups occur as regular Galois groups over Q ( t ) and so also over Q . We show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0. [ABSTRACT FROM AUTHOR]

Abstract: For any number field K, it is unknown which finite groups appear as Galois groups of extensions such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For , the answer is described by the long standing -admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tamely ramified maximal subfields of K-division algebras, generalizing Liedahlʼs theorem for metacyclic G and Sonnʼs solution of the -admissibility conjecture for solvable groups. [Copyright &y& Elsevier]

Abstract: Let G be a finite group, K be a field and be a faithful representation where V is a finite-dimensional vector space over K. In this paper, we will consider the problem which asks whether the fixed field is rational (i.e. purely transcendental) over K. This is a variant of Noether''s problem. We will show the complete results of this problem in the case where G is a 2-group, and and 5. Our results imply the existence of a generic polynomial for the corresponding group. [Copyright &y& Elsevier]