This section sets up many of the basic notions used in this book.
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
![{\displaystyle {\frac {\mathbb {F} _{p}[x]}{(x^{2}-a)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29c3a022f4236cdb1af691d9100112d0d1d1cc36)
is a field or not. For example,
has no solutions in
while
has
as solutions. This implies that
while ![{\displaystyle {\frac {\mathbb {F} _{5}[x]}{(x^{2}+1)}}\cong \mathbb {F} _{5}\times \mathbb {F} _{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a96245bf871f5d3b53111201c14548a81294956c)
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.
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
![{\displaystyle x^{q-1}\equiv 1{\text{ }}({\text{mod }}p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb6ba6a12c0891a0b86c4b0c019925bfce87aa1)
Note Fermat's original theorem proved the case
.
The most useful techniques used in this section are the applications of the Euler
-function.
This section studies sets of the form
![{\displaystyle \{p\in \mathbb {F} _{q}^{n}:f_{1}(p)=\cdots =f_{k}(p)=0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a006d33a553b156fc9efc4966fcdfa6bf70ac88)
where
. If you are used to scheme theory, Serre studies schemes of the form
![{\displaystyle X={\text{Spec}}\left({\frac {\mathbb {Z} [x_{1},\ldots ,x_{n}]}{(f_{1},\ldots ,f_{k})}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/227e26d64688ce99bcfa3b49e6999278e3804951)
by looking at the sets
![{\displaystyle X(\mathbb {F} _{q})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9058dd42114a5a0f7bc688ceab0cb5302dc70ba6)
This section introduces a technical tool for proving the Chevalley-Warning theorem. It relies on the following
![{\displaystyle 0+1+2+\cdots +p=(0+p)+(1+(p-1))+(2+(p-2))+\cdots =k\cdot p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/578efa17812d2869237017ae4bf31aab9f06dd23)
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 ![{\displaystyle 0({\text{mod }}p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61ba8ef5ba8bf499d65c5f8819c95c4bf03d67a)
The most interesting technical tool used in the proof of this theorem is the indicator function
which could be equivalently described as the function
![{\displaystyle P(x)={\begin{cases}1&{\text{ if }}x\in V\\0&{\text{ otherwise}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54065cab25b362e089be15f7c3331a97e5cba5b5)
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.
This section gives us the construction of the Legendre symbol and Gauss' reciprocity theorem.
The theorem is the setup for the definition of the Legendre symbol, which is defined as the second map in the short exact sequence
![{\displaystyle 1\to \mathbb {F} _{q}^{*2}\to \mathbb {F} _{q}^{*}\to \{\pm 1\}\to 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91d54eee6dcde0f02e862d4b37fee2cd6521ce3c)
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
![{\displaystyle (\mathbb {F} _{9}^{*})^{2}=\{1,2,i,2i\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38ca7bbd4a5786375c49777c0831372067fd8acd)
hence
![{\displaystyle \mathbb {F} _{3^{3}}=\mathbb {F} _{27}\cong \mathbb {F} _{9}[y]/(y^{2}+2i+2)\cong F_{3}[x,y]/(x^{2}+1,y^{2}+x+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ed2b50fee1f771b93d526957741db18b86ea26)
Here Serre restricts to the classical case of the sequence
![{\displaystyle 1\to \mathbb {F} _{p}^{*2}\to \mathbb {F} _{p}^{*}\to \{\pm 1\}\to 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7a036f70ea09ea85063cdedb487871782ff0e7)
and defines the second map as the Legendre symbol
![{\displaystyle \left({\frac {\cdot }{p}}\right):\mathbb {F} _{p}^{*}\to \mathbb {Z} /2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c435ad028ca5033a1b695c804ef7fcea96ab644e)
If you embed
, then the Legendre symbol is an example of a character (a group morphism
). This means that
![{\displaystyle \left({\frac {ab}{p}}\right)=\left({\frac {a}{p}}\right)\cdot \left({\frac {b}{p}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0469f63db8f1fe7d4a1d38f47d832bd2837307)
In addition, the Legendre symbol can be extended to
by setting ![{\displaystyle {\frac {0}{p}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cda39d03c7b7260a3d984194f6100f2bfda32ec8)
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
:
![{\displaystyle \varepsilon (n)\equiv {\frac {n-1}{2}}{\text{ }}({\text{mod }}4)={\begin{cases}0&{\text{ if }}n\equiv 1{\text{ }}({\text{mod }}4)\\1&{\text{ if }}n\equiv -1{\text{ }}({\text{mod }}4)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd294e1577496705383ac28c54cd033461f419c)
![{\displaystyle \omega (n)\equiv {\frac {n^{2}-1}{8}}{\text{ }}({\text{mod }}8)={\begin{cases}0&{\text{ if }}n\equiv \pm 1{\text{ }}({\text{mod }}8)\\1&{\text{ if }}n\equiv \pm 5{\text{ }}({\text{mod }}8)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfe98fca44060d8cac7dfb4241560f95d8c43e3)
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
![{\displaystyle \left({\frac {-1}{p}}\right)=\left({\frac {p-1}{p}}\right)=(-1)^{\varepsilon (p)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60b90a1450fa1df323cafbbfa255fc602ae305ca)
If we split
into the two cases of odd numbers, then
![{\displaystyle p-1={\begin{cases}(4k+1)-1=4k\\(4k+3)-1=4k+2\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80290f636f680ed6f8f909fbcc88caf31d620909)
Then, using the definition of the Legendre symbol, we find that
![{\displaystyle \left({\frac {-1}{p}}\right)=(-1)^{\frac {p-1}{2}}={\begin{cases}(-1)^{\frac {4k}{2}}=(-1)^{2k}=1&{\text{if }}p\equiv 1{\text{ }}({\text{mod }}4)\\(-1)^{\frac {4k+2}{2}}=(-1)^{2k+1}=-1&{\text{if }}p\equiv 3{\text{ }}({\text{mod }}4)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f05d7a41907b3d73acb3e99ee8d02643e1ce190d)
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
![{\displaystyle 8k+1,8k+3,8k+5,8k+7}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f494306c3cbc66e9ce305ccfcd54c53d02b429)
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 ![{\displaystyle \zeta _{8}^{4}\cdot \zeta _{8}^{-4}=(-1)\cdot \zeta _{8}^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01dbeaa0d5d8239b01d39d82ff7bcde7c942d4ce)
implying that
forcing ![{\displaystyle \zeta _{8}^{2}+\zeta _{8}^{-2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbb74d0c9d4e808a1463f051f6b6c90f9ebf82e)
hence
satisfies
since
![{\displaystyle y^{2}=(\zeta _{8}+\zeta _{8}^{-1})^{2}=\zeta _{8}^{2}+2+\zeta _{8}^{-2}=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c258a009934917d5cbc1b1da8a9773cb21f7f40)
Since the Frobenius
is an automorphism of
, we have that
![{\displaystyle y^{p}=\zeta _{8}^{p}+\zeta _{8}^{-p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c91e2cf3be27240c1629b74237ef660732955dd)
If
then
. This implies
![{\displaystyle \left({\frac {2}{p}}\right)=y^{p-1}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfbbe8dc0d026db4dd53de4f8b154d385c2622ea)
Otherwise, if
then
(draw a picture of the unit circle to check that
and
). Hence
.
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.
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
![{\displaystyle {\frac {\mathbb {F} _{43}[x]}{(x^{2}-29)}}\cong \mathbb {F} _{1849}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbdeff7eb27aa598c1713fc34b63719dcbf7b51a)
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:
![{\displaystyle \left({\frac {l}{p}}\right)=(-1)^{\varepsilon (p)\varepsilon (l)}\left({\frac {p}{l}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d96b1665091dc1eca7f1506335c624c37bafac6)
His sample computation goes as follows:
![{\displaystyle {\begin{aligned}\left({\frac {29}{43}}\right)&=\left({\frac {43}{29}}\right)&{\text{ since }}\varepsilon (43)=0\\&=\left({\frac {14}{29}}\right)&{\text{ since }}[43]=[14]\\&=\left({\frac {2}{29}}\right)\cdot \left({\frac {7}{29}}\right)&{\text{ from being a group morphism }}\\&=-\left({\frac {7}{29}}\right)&{\text{ since }}29\equiv 5{\text{ }}({\text{mod }}8)\\&=-\left({\frac {29}{7}}\right)&{\text{ since }}\varepsilon (7)=0\\&=-\left({\frac {1}{7}}\right)=-1&\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb08cf747870414febab86d58c3ba0185710cc25)
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.