# A User's Guide to Serre's Arithmetic

This wikibook is a companion guide to Serre's book on arithmetic. His proofs will be dissected, external references will be made, technique discussed, and computations made

## Finite Fields

In this chapter Serre studies the basic properties of finite fields. Of them, he gives uniqueness of finite fields ${\displaystyle \mathbb {F} _{q}}$  of characteristic ${\displaystyle p}$  and order ${\displaystyle q=p^{f}}$ , gives the structure of the group ${\displaystyle \mathbb {F} _{q}^{*}}$ , discusses solutions of polynomials over finite fields, and gives a necessary and sufficient condition for

${\displaystyle {\frac {\mathbb {F} _{q}[x]}{(x^{2}-a)}}}$

to be a field extension or the product ring ${\displaystyle \mathbb {F} _{q}\times \mathbb {F} _{q}}$ .