Artin–Schreier theory

In mathematics, Artin–Schreier theory is a branch of Galois theory, and more specifically is a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p. Artin and Schreier (1927) introduced Artin–Schreier theory for extensions of prime degree p, and Witt (1936) generalized it to extensions of prime power degree pⁿ.

If K is a field of characteristic p, a prime number, any polynomial of the form

for ${\displaystyle \alpha }$ in K, is called an Artin–Schreier polynomial. When ${\displaystyle \alpha }$ does not lie in the subset ${\displaystyle \{y\in K\,|\,y=x^{p}-x\;{\mbox{for }}x\in K\}}$, this polynomial is irreducible in K[X], and its splitting field over K is a cyclic extension of K of degree p. This follows since for any root β, the numbers β + i, for ${\displaystyle 1\leq i\leq p}$, form all the roots—by Fermat's little theorem—so the splitting field is ${\displaystyle K(\beta )}$.

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