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 ${𝔽}_q$ of characteristic $p$ and order $q=p^f$, gives the structure of the group ${𝔽}_q^{*}$, discusses solutions of polynomials over finite fields, and gives a necessary and sufficient condition for

$\frac{{𝔽}_q[x]}{(x^2-a)}$

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

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/