Toward quantization of Galois theory

Toward Nevanlinna-galois Theory for Algebraic Minimal Surfaces

A complete minimal surface in R is said to be pseudo-algebraic, if its Weierstrass data are defined on a compact Riemann surface finitely many points removed M = M\{P1, . . . , Pn} and extend meromorphically across punctures ([KKM], see also [G]). The punctured Riemann surface M on which the Weierstrass data are defined is called the basic domain. The difference between a complete minimal surfa...

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