Fields form one of the basic algebraic concepts, for which there is an extensive theory, dealing mainly with the form taken by field extensions. In Section 7.1 field extensions are described, and the special case of splitting fields is introduced in Section 7.2, leading to the notion of algebraic closure (Section 7.3). In Section 7.4 we examine the problems arising in finite characteristic. One of the main tools in this study is Galois theory and this forms the subject of Sections 7.5 and 7.6, while Sections 7.10 and 7.11 bring its application to the solution of equations . The special case of finite fields is studied in Section 7.8, using information on the roots of unity (Section 7.7); Section 7.9 is devoted to generators and some invariants of extensions.
KeywordsFinite Field Galois Group Minimal Polynomial Galois Theory Galois Extension
Unable to display preview. Download preview PDF.