Artin-Schreier Theory

Artin-Schreier theory studies certain extensions of fields, particularly those with characteristic p (a prime number). It describes how field extensions can be constructed by adding solutions to equations of the form \( x^p - x = a \), where \( a \) is in the base field. These extensions are called Artin-Schreier extensions and are analogous to quadratic extensions in characteristic not p. The theory classifies all such extensions, linking them to additive polynomials and revealing their structure as cyclic, degree p, Galois extensions. It plays a key role in understanding the algebraic structure of fields with positive characteristic.