Automorphism groups over Hilbertian fields

HILBERTIAN FIELDS AND FREE PROFINITE GROUPS* by

Automorphism Groups of Simple Moufang Loops over Perfect Fields

Let F be a perfect field and M(F ) the nonassociative simple Moufang loop consisting of the units in the (unique) split octonion algebra O(F ) modulo the center. Then Aut(M(F )) is equal to G2(F )o Aut(F ). In particular, every automorphism of M(F ) is induced by a semilinear automorphism of O(F ). The proof combines results and methods from geometrical loop theory, groups of Lie type and compo...

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 ...

Automorphism Towers and Automorphism Groups of Fields without Choice

Orbits of Automorphism Groups of Fields

This paper groups together some results that share a theme (orbits of automorphism groups on elements of a field) and some proof ideas (constructing an additive or multiplicative “trace/norm map”). Namely, we prove that a field whose automorphism group acts with finitely many orbits must be finite (Theorem 1.1), we discuss a stronger conjecture, and we show that the automorphisms of a Mal’cev-N...