Let $L/F$ be an unramified extension of $p$-adic fields. Then it is Galois and $Gal(L/F)$ is cyclic, generated by an element $\sigma$ called the Frobenius element. Now, in the book I'm reading, the author states "The latter is characterized by the congruence $\sigma(x)\equiv x^q\mod\mathfrak{P}$ for all $x\in\mathcal{O}_L$, where $\mathfrak{P}$ is the maximal ideal of $\mathcal{O}_L$ and $q$ is the size of the residue field $\mathbb{L}=\mathcal{O}_L/\mathfrak{P}$."
I know that since the extension is unramified that $Gal(L/F)\cong Gal(\mathbb{L}/\mathbb{F})$ where $\mathbb{F}$ is the residue field for $F$. My question is shouldn't we have $p$ instead of $q$ in the above paragraph defining the Frobenius element? Don't we have $x^q=x$ in general?