Probabilistic Galois theory

Galois Lattice Theory for Probabilistic Visual Landmarks

This paper presents an original application of the Galois lattice theory, the visual landmark selection for topological localization of an autonomous mobile robot, equipped with a color camera. First, visual landmarks have to be selected in order to characterize a structural environment. Second, such landmarks have to be detected and updated for localization. These landmarks are combinations of...

Galois Theory

Galois Theory

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...

Galois Theory

Proposition 1.3. Let φ be an automorphism of a field extension K/F , and f(x) ∈ F [x]. Let α1, . . . , αn be the roots of f(x) lying in K. Then φ permutes the set {α1, . . . , αn}. If also the set of αi generate K over F , then two automorphisms φ1, φ2 of K/F which agree on all the αi are equal. Thus, in this case we have an inclusion of Aut(K/F ) as a subgroup of Sym({α1, . . . , αn}) ∼= Sn. P...

Galois Theory