Frobenius Endomorphism - Fixed Points of The Frobenius Endomorphism

Fixed Points of The Frobenius Endomorphism

Say R is an integral domain. The Frobenius map fixes all the elements of R which satisfy the equation xp = x. These are all the roots of the equation xp - x, and since this equation has degree p, there are at most p roots. These are exactly the elements 0, 1, 2, ..., p - 1, so the fixed point set of F is the prime field.

Iterating the Frobenius map gives us a sequence of elements in R:

Applying the e'th iterate of F to a ring which contains a field K of pe elements gives us a fixed point set equal to K, similar to the example above. The iterates of the Frobenius map are also used in defining the Frobenius closure and tight closure of an ideal.