Title of the course: Finite Fields
Instructor: Dr. José-Ibrahim Villanueva-Gutiérrez
Institution: Universität Heidelberg
Dates: 16-22 September 2019
Prerequisites: Linear algebra, Algebra I
Level: Beginners
Abstract: Finite fields are present in many areas of mathematics. They frequently appear in number theory: the residue field of non-archimedian local fields are precisely finite fields. Finite fields provide a very mild environment to work on. For instance, the extensions of degree pn of a finite field of characteristic p, are pairwise isomorphic. Finite fields belong to the part of the nice puzzle of mathematics, where all pieces fall straightforward.
In this course we will study the main algebraic properties of finite fields. In particular, we will study their group of automorphisms. More generally we will study the group of automorphisms of the algebraic closure Fp of a finite field Fp. To this end we will dig in the topological Galois theory. We will show that the group of such automorphisms is isomorphic to the pro-finite completion bZ of Z.
Language: EN
Textbook: Short notes and exercices to be published in https://www.mathi.uni-heidelberg.de/~jgutierrez
Other references:
Serge Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002.
Robert B. Ash. Basic abstract algebra. Dover Publications, Inc., Mineola, NY, 2007. For graduate students and advanced undergraduates.