The Embedding Problem Over a Hilbertian PAC-Field

The Embedding Problem over a Hilbertian Pac-field

We show that the absolute Galois group of a countable Hilbertian P(seudo)A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G(Q̄/Q) is the extension of groups with a fairly simple structure (e.g., ∏∞ n=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a...

On hermitian trace forms over hilbertian fields

Let k be a field of characteristic different from 2. Let E/k be a finite separable extension with a k-linear involution σ. For every σ-symmetric element μ ∈ E∗, we define a hermitian scaled trace form by x ∈ E 7→ TrE/k(μxx). If μ = 1, it is called a hermitian trace form. In the following, we show that every even-dimensional quadratic form over a hilbertian field, which is not isomorphic to the ...

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...

Pac Fields over Finitely

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K.

INTERSECTIONS OF LOCAL ALGEBRAIC EXTENSIONS OF A HILBERTIAN FIELD* by