A User's Guide to Serre's Arithmetic/Finite Fields

GeneralitiesEdit

This section sets up many of the basic notions used in this book.

Finite FieldsEdit

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

  while  

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.

Multiplicative Group of a Finite FieldEdit

This section is dedicating to show that the multiplicative group   is cyclic of order  . He does this through proving a stronger result that all subgroups of   are cyclic.

In addition, while proving the theorem, he shows a generalization of Fermat's Little Theorem which states

 

Note Fermat's original theorem proved the case  .

The most useful techniques used in this section are the applications of the Euler  -function.

Equations Over a Finite FieldEdit

This section studies sets of the form

 

where  . If you are used to scheme theory, Serre studies schemes of the form

 

by looking at the sets

 

Power SumsEdit

This section introduces a technical tool for proving the Chevalley-Warning theorem. It relies on the following

  1.  

Chevalley TheoremEdit

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.

Quadratic Reciprocity LawEdit

This section gives us the construction of the Legendre symbol and Gauss' reciprocity theorem.

Squares in  Edit

The theorem is the setup for the definition of the Legendre symbol, which is defined as the second map in the short exact sequence

 

This morphism is defined by using the generalization of Fermat's little theorem. Since   we have that  . For an application of this sequence recall that  . We can calculate that

 

hence

 

Legendre Symbol (Elementary Case)Edit

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

  by setting  

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

 

as desired. 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  

implying that

  forcing  

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  .

Small RemarkEdit

Furthermore, using the fact that the Legendre symbol is a group morphism, we can compute the Legendre symbol of   many elements for   without having to compute explicit squares.

Quadratic Reciprocity LawEdit

This section is dedicated to proving Quadratic reciprocity. As we have said before, this is a useful computational tool for determining if

  is a field extension

He gives a computation of the Legendre symbol to determine that

 

I will simply state quadratic reciprocity and give references to other proofs.

Theorem: Given a pair of distinct odd prime number  ,  we have the following reciprocity law:

 

His sample computation goes as follows:

 

There are nice discussions about the proofs of quadratic reciprocity on mathoverflow

and here is a compilation of hundres of proofs for quadratic reciprocity

Try reading the proof https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity#Proof_using_algebraic_number_theory to motivate the generalization to Artin reciprocity.