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

p-adic Fields

Hilbert Symbol

Quadratic Forms Over Q p and Q

Integral Quadratic Forms with Discriminant ±1

The Theorem on Arithmetic Progressions

Modular Forms

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/