A User's Guide to Serre's Arithmetic

G

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 $\mathbb {F} _{q}\times \mathbb {F} _{q}$.

References

• Serre - A Course in Arithmetic
• https://people.ucsc.edu/~weissman/Math222A/SerreAnn.pdf
• https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/