This chapter starts out with a discussion of the structure of finite fields. Given a field its characteristic is defined as the smallest number such that is congruent to zero in . If this number is unbounded, then we say is of characteristic 0. This is well-defined because every ring has a unique morphism .
For a field of positive characteristic, denoted where , he goes on to show that for some prime and some integer and that the characteristic of such a field is .
Before stating this theorem he proves a lemma showing that the frobenius
given by is an injective morphism onto a subfield of , (FIX: the subfield of numbers in invariant under...). This can be used to show that for an algebraic closure of , is an automorphism.
The theorem also states that all finite fields of order are isomorphic to . It's worth noting that the technique of looking at a polynomial and its derivative is a common and useful technical tool.
The question you should be asking yourself is:
How can I construct finite fields of order greater than ?
We can use the case of the real numbers to get a hint: we should look at quadratic polynomial and see if
is a field or not. For example, has no solutions in while has as solutions. This implies that
The rest of the chapter is dedicated to building tools for determining if a quadratic function determines a field extension of a finite field. Note that this will give us a recursive method for finding any . In addition, we will construct a tool and a theorem, called the Legendre Symbol and Gauss' reciprocity theorem, for efficiently figuring out if determines a field extension or product of fields.
The Chevallay-Warning theorem gives a useful criterion for determining the number of solutions to a set of polynomials over a finite field. I will restate it here for convenience
Given polynomials such that . The cardinality of is congruent to
The most interesting technical tool used in the proof of this theorem is the indicator function which could be equivalently described as the function
Note that the -power is an application of the generalization of Fermat's little theorem proved in the last section.
This theorem has numerous applications. First, it solves many arithmetic questions about the existence of solutions of polynomials over finite fields. This is stated in corollary 1. Also, he shows that a for a quadratic form (meaning the give a symmetric matrix) has a non-zero solution over every finite field.
Here Serre restricts to the classical case of the sequence
and defines the second map as the Legendre symbol
If you embed , then the Legendre symbol is an example of a character (a group morphism ). This means that
In addition, the Legendre symbol can be extended to
Notice that we can lift the Legendre symbol to using the composition of the quotient map with the Legendre symbol.
Finally, he finds a method for computing the Legendre symbol of . The first case is easy since . For the last two cases he introduces a couple auxillary functions from the odd integers to :
Recall from elementary number theory that every odd number greater than is of the for or (they can't be of the form or since those are even). Then, acts as a function partitioning off the two sets of odd numbers. In addition, there are infinitely many prime numbers in both forms. Serre claims that
If we split into the two cases of odd numbers, then
Then, using the definition of the Legendre symbol, we find that
In the last case, Serre again uses a function which partitions off the odd numbers. Notice that every odd number (hence ever prime greater then ) is of one of the forms
In order to take advantage of this partition, he embeds and claims that where is the primitive -th root of unity (in the complex numbers . This follows from the observation that
hence since and
hence satisfies since
Since the Frobenius is an automorphism of , we have that
If then . This implies
Otherwise, if then (draw a picture of the unit circle to check that and ). Hence .