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Published in:
2018 | OriginalPaper | Chapter
4. Algebraic Extensions
Author : Juliusz Brzeziński
Published in: Galois Theory Through Exercises
Publisher: Springer International Publishing
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Abstract
This chapter describes a kind of field extension most common in Galois theory: algebraic extensions, i.e. extensions K ⊆ L in which each element α ∈ L is a zero of a nontrivial polynomial with coefficients in K. We relate the elements of algebraic extensions to the corresponding polynomials and look at the structures of the simplest extensions K(α) of K. We also introduce the notion of the degree of a field extension and prove some of its properties such as the Tower Law.