In attempting to solve the regular inverse Galois problem for arbitrary subfields K of C (particularly for K=Q), a very important result by Fried and Völklein reduces the existence of regular Galois extensions F|K(t) with Galois group G to the existence of K-rational points on components of certain moduli spaces for families of covers of the projective line, known as Hurwitz spaces. In some cases, the existence of rational points on Hurwitz spaces has been proven by theoretical criteria. In general, however, the question whether a given Hurwitz space has any rational point remains a very difficult problem. In concrete cases, it may be tackled by an explicit computation of a Hurwitz space and the corresponding family of covers. The aim of this work is to collect and expand on the various techniques that may be used to solve such computational problems and apply them to tackle several families of Galois theoretic interest. In particular, in Chapter 5, we compute explicit curve equations for Hurwitz spaces for certain families of \(M_{24}\) and \(M_{23}\). These are (to my knowledge) the first examples of explicitly computed Hurwitz spaces of such high genus. They might be used to realize \(M_{23}\) as a regular Galois group over Q if one manages to find suitable points on them. Apart from the calculation of explicit algebraic equations, we produce complex approximations for polynomials with genus zero ramification of several different ramification types in \(M_{24}\) and \(M_{23}\). These may be used as starting points for similar computations. The main motivation for these computations is the fact that \(M_{23}\) is currently the only remaining sporadic group that is not known to occur as a Galois group over Q. We also compute the first explicit polynomials with Galois groups \(G=P\Gamma L_3(4), PGL_3(4), PSL_3(4)\) and \(PSL_5(2)\) over Q(t). Special attention will be given to reality questions. As an application we compute the first examples of totally real polynomials with Galois groups \(PGL_2(11)\) and \(PSL_3(3)\) over Q. As a suggestion for further research, we describe an explicit algorithmic version of "Algebraic Patching", following the theory described e.g. by M. Jarden. This could be used to conquer some problems regarding families of covers of genus g>0. Finally, we present explicit Magma implementations for several of the most important algorithms involved in our computations. / Das Umkehrproblem der Galoistheorie und explizite Berechnung von Familien von Überlagerungen des \(\mathbb{P}^1\mathbb{C}\) mit vorgegebener Verzweigung

Mikroprimstellen wurden eingeführt von J. Neukirch im Rahmen der abstrakten Klassenkörpertheorie. Eine Verallgemeinerung der Zerlegungsgruppen von Primstellen globaler Körper motivierte die rein gruppentheoretische Definition der Mikroprimstellen als gewisse Äquivalenzklassen von Frobeniuselementen. Auf den Fall der Galoisgruppen lokaler oder globaler Körper angewendet, ergibt diese Theorie eine Beschreibung spezieller Konjugationsklassen. Die Hauptaufgabe von J. Neukirch ist, die zahlentheoretische Bedeutung der Mikroprimstellen zu verstehen, das heißt, sie in Termen des Grundkörpers anzugeben. J. Mehlig und E.-W. Zink fanden eine Bijektion zwischen Mikroprimstellen und normverträglichen Folgen von Primelementen in Körpertürmen. Diese Türme entstehen durch die Fixkörper der abgeleiteten Untergruppen der Trägheitsgruppe. Auf diese Weise betrachtet man Mikroprimstellen für die entsprechenden Faktorgruppen der absoluten Galoisgruppe, um dann einen projektive Limes zu bilden. Im ersten Schritt ist eine Bijektion zwischen relativen Mikroprimstellen und Konjugationsklassen von Primelementen gezeigt worden. Das Hauptergebnis dieser Arbeit ist eine vollständige Antwort auf die Frage von J. Neukirch im zweiten Schritt. Es wird eine Normabbildung für Lubin-Tate-Potenzreihen verschiedener Höhe angegeben und der projektive Limes bezüglich dieser Normabbildungen gebildet. Dazu werden Ergebnisse der Klassenkörpertheorie auf einen ''''fastabelschen'''' Fall übertragen. Schließlich können die Mikroprimstellen als Galoisorbits von normverträglichen Abfolgen normischer Lubin-Tate-Potenzreihen beschrieben werden. Die Koeffizienten aller dieser Lubin-Tate-Potenzreihen sind in einer endlichen unverzweigten Erweiterung des Grundkörpers. Also kann man zu einer gegebenen normverträglichen Abfolge normischer Lubin-Tate-Potenzreihen den Koeffizientenkörper definieren. Der Grad dieses Körpers bzw. die Länge des Galoisorbits entspricht dem Grad der zugehörigen Mikroprimstelle. / Micro primes were introduced by J. Neukirch in the context of abstract class field theory. A generalization of decomposition groups of primes of global fields led him to a purely group theoretical definition of micro primes as certain equivalence classes of Frobenius elements. Applied to the case of Galois groups of local or global fields this theory yields a description of special conjugacy classes. The main problem already posed by J. Neukirch is to understand the number theoretical meaning of micro primes, that is to describe them in terms of the base field. J. Mehlig and E.-W. Zink established a bijection between micro primes and norm compatible sequences of prime elements in field towers. These towers arise as fixed point fields for the sequence of derived subgroups of the inertia group. So one has to study micro primes for the corresponding factor groups of the absolute Galois group and then to form a projective limit. In the first step, a bijection between relative micro primes and conjugacy classes of prime elements has been obtained. The main result of this project is a complete answer to the problem of J. Neukirch for the second step. One has to introduce norm maps between Lubin-Tate power series of different height and the projective limit has to be taken with respect to these norm maps. For this purpose results from class field theory are transferred to an ''''almost abelian'''' case. In the end micro primes can be described as Galois orbits of norm compatible sequences of normic Lubin-Tate power series. The coefficients of all the Lubin-Tate power series are in finite unramified extensions of the base field. Therefore one can define a field of coefficients for a given norm compatible sequence of normic Lubin-Tate power series. The degree of that field respectively the length of the Galois orbit is at the same time the degree of the corresponding micro prime.