Notes on Galois Theory II

ثبت نشده

چکیده

Lemma 2.1. Let F be a field, let E = F (α) be a simple extension of F , where α is algebraic over F and f = irr(α, F, x), let ψ : F → K be a homomorphism from F to a field K, and let L be an extension of K. If β ∈ L is a root of ψ(f), then there is a unique extension of ψ to a homomorphism φ : E → L such that φ(α) = β. Hence there is a bijection from the set of homomorphisms φ : E → L such that φ(a) = ψ(a) for all a ∈ F to the set of roots of the polynomial ψ(f) in L, where ψ(f) ∈ K[x] is the polynomial obtained by applying the homomorphism ψ to coefficients of f .

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Deformation of Outer Representations of Galois Group II

This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for...

متن کامل

A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...

متن کامل

Algebraic and Algorithmic Aspects of Difference Equations∗

In this course, I will give an elementary introduction to the Galois theory of linear difference equations. This theory shows how to associate a group of matrices with a linear difference equation and shows how group theory can be used to determine properties of the solutions of the equations. I will begin by giving an introduction to the theory of linear algebraic groups, those groups that occ...

متن کامل

Math 121 Notes: a Field Guide to Galois Theory

1. Vector Spaces: 1/7/13 1 2. Review of Math 120 I: Planting Seeds: 1/9/13 3 3. Review of Math 120 II: Harvesting Consequences: 1/11/13 4 4. Extending Fields by Adjoining Roots of Irreducible Polynomials: 1/14/13 5 5. Splitting Fields and Algebraic Closures: 1/16/13 6 6. Searching for Closure: 1/18/13 7 7. Separable Field Extensions: 1/23/13 9 8. Finite Fields and Roots of Unity: 1/25/13 10 9. ...

متن کامل