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 $$\mathbb{F}_q$$ of characteristic $$p$$ and order $$q=p^f$$, gives the structure of the group $$\mathbb{F}_q^*$$, discusses solutions of polynomials over finite fields, and gives a necessary and sufficient condition for
 * $$\frac{\mathbb{F}_q[x]}{(x^2 - a)}$$

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