# Prime Power Mapping on Galois Field is Automorphism

## Theorem

Let $\GF$ be a Galois field whose zero is $0_\GF$ and whose characteristic is $p$.

Let $\sigma: \GF \to \GF$ be defined as:

$\forall x \in \GF: \map \sigma x = x^p$

Then $\sigma$ is an automorphism of $\GF$.

## Proof

Let $x, y \in \GF$.

Then:

 $\ds \map \sigma {x y}$ $=$ $\ds \paren {x y}^p$ Definition of $\sigma$ $\ds$ $=$ $\ds x^p y^p$ Power of Product of Commutative Elements in Group $\ds$ $=$ $\ds \map \sigma x \map \sigma y$ Definition of $\sigma$

and:

 $\ds \map \sigma {x + y}$ $=$ $\ds \paren {x + y}^p$ Definition of $\sigma$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^p \dbinom p k x^k y^{p - k}$ Binomial Theorem $\ds$ $\equiv$ $\ds x^p + y^p$ $\ds \pmod p$ Power of Sum Modulo Prime $\ds$ $=$ $\ds \map \sigma x + \map \sigma y$ Definition of $\sigma$

Thus it has been demonstrated that $\sigma$ is a homomorphism.

Then we have:

 $\ds \map \ker \sigma$ $=$ $\ds \set {x \in \GF: \map \sigma x = 0_\GF}$ Definition of Kernel of Ring Homomorphism $\ds$ $=$ $\ds \set {x \in \GF: x^p = 0_\GF}$ Definition of $\sigma$ $\ds$ $=$ $\ds \set {x \in \GF: x = 0_\GF}$ Congruence of Powers $\ds$ $=$ $\ds \set {0_\GF}$ Congruence of Powers

From Kernel is Trivial iff Monomorphism, $\sigma$ is a monomorphism.

That is, $\sigma$ is an injection.

Then from Injection from Finite Set to Itself is Surjection, $\sigma$ is a surjection.

Thus $\sigma$ is a bijective homomorphism to itself.

The result follows by definition of automorphism.

$\blacksquare$