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

## Generalities

[edit | edit source]This section sets up many of the basic notions used in this book.

### Finite Fields

[edit | edit source]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 Field

[edit | edit source]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 Field

[edit | edit source]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 Sums

[edit | edit source]This section introduces a technical tool for proving the Chevalley-Warning theorem. It relies on the following

### Chevalley Theorem

[edit | edit source]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 Law

[edit | edit source]This section gives us the construction of the Legendre symbol and Gauss' reciprocity theorem.

### Squares in

[edit | edit source]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 | edit source]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 Remark

[edit | edit source]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 Law

[edit | edit source]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 hundreds 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**.