Fermat's Little Theorem, A categorical approach

Proposition. Let ${\mathrm{F}}_{(p)}$$\mathrm{F}_{(p)}$ be the category of fields with $\mathrm{char}F=p$$\mathrm{char} F = p$, then $\mathbb{Z}/p\mathbb{Z}$$\mathbb{Z}/p\mathbb{Z}$ is the initial object.

Proof. Since $\mathbb{Z}$$\mathbb{Z}$ is the initial object in $\mathrm{Ring}$$\mathrm{Ring}$, $\mathrm{char};F=p$$\mathrm{char}; F = p$ shows that the image of the unique map $\iota :\mathbb{Z}\to F$$\iota: \mathbb{Z} \to F$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$$\mathbb{Z}/p\mathbb{Z}$. Then $\phi :\mathbb{Z}/p\mathbb{Z}\to F,1_p\mapsto 1_F$$\varphi: \mathbb{Z}/p\mathbb{Z} \to F, 1_p \mapsto 1_F$ is the unique map. $◻$$\square$

Corollary: Fermat's Little Theorem.

Observe that $f:a\mapsto a^p$$f: a \mapsto a^p$ is an endomorphism of $\mathbb{Z}/p\mathbb{Z}$$\mathbb{Z}/p\mathbb{Z}$ since

$$\begin{array}{}\text{(1)}& (a+b{)}^p=a^p+(\genfrac{}{}{0ex}{}{p}{1})a{b}^{p-1}+...+(\genfrac{}{}{0ex}{}{p}{p-1})a^{p-1}b+b^p=a^p+b^p,(ab{)}^p=a^p{b}^p,1^p=1.\end{array}$$

but since $\mathbb{Z}/p\mathbb{Z}$$\mathbb{Z}/p\mathbb{Z}$ is the initial object, there is only one endomorphism on $\mathbb{Z}/p\mathbb{Z}$$\mathbb{Z}/p\mathbb{Z}$, that is, $\mathrm{id}$$\mathrm{id}$. Hence $f(x)=x^p=x$$f(x) = x^p = x$ in $\mathbb{Z}/p\mathbb{Z}$$\mathbb{Z}/p\mathbb{Z}$.

In general, Let $R_{(p)}$$R_{(p)}$ be the ring such that $\mathrm{char}R=p$$\mathrm{char} R=p$, ${\mathbb{F}}_p$$\mathbb F_p$ is the initial object of $R_{(p)}$$R_{(p)}$. Let $\phi :{\mathbb{F}}_p\to R$$\varphi:\mathbb F_p\to R$ be the unique map.

Then $f:R\to R,r⟼r^p$$f:R\to R,r\longmapsto r^p$ is a ring endomorphism for char $R=p$$R=p$, then we have $f\circ \phi :{\mathbb{F}}_p\to R=\phi$$f\circ \varphi:\mathbb F_p\to R=\varphi$ since $\phi$$\varphi$ is the unique morphism.