Random Galois extensions of Hilbertian fields

Galois Extensions of Hilbertian Fields

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all ∈ G(K)e, the field Ks[ ] ∩Ktot,S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, K̂p. Then Kp = Ks ∩ K̂p and Ktot,S = T p∈S T σ∈G(K) K σ p . G(K) stands for the absolute Galois ...

Compositum of Galois Extensions of Hilbertian Fields

Large Normal Extensions of Hilbertian Fields

Let K be a countable separably Hilbertian field. Denote the absolute Galois group of K by G(K). For each σ ∈ (σ1, . . . , σe) ∈ G(K) let Ks[σ] be the maximal Galois extension of K which is fixed by σ1, . . . , σe. We prove that for almost all σ ∈ G(K) (in the sense of the Haar measure) the field Ks[σ] is PAC and its absolute Galois group is isomorphic to F̂ω. Mathematische Zeitschrift 224 (1997)...

The Inverse Galois Problem, Hilbertian Fields, and Hilbert’s Irreducibility Theorem

In the study of Galois theory, after computing a few Galois groups of a given field, it is very natural to ask the question of whether or not every finite group can appear as a Galois group for that particular field. This question was first studied in depth by David Hilbert, and since then it has become known as the Inverse Galois Problem. It is usually posed as which groups appear as Galois ex...

Procyclic Galois Extensions of Algebraic Number Fields

6 1 Iwasawa’s theory of Zp-extensions 9 1.