Finite fields

In this subsection we study some basic properties on finite fields. A good account can be found in [2]. Also, there is a brief explanation in [1] available on the net.

LEMMA 0.3   Let $F$ be a finite field (that means, a field which has only a finite number of elements.) Then:

1. There exists a prime number $p$ such that $p=0$ holds in $F$.
2. $F$ contains $\mathbb{F}_p$ as a subfield.
3. $q=\char93 (F)$ is a power of $p$.
4. For any $x\in F$, we have $x^q-x=0$.
5. The multiplicative group $(F_q)^{\times}$ is a cyclic group of order $q-1$.

The next task is to construct such fields. An important tool is the following lemma.

LEMMA 0.4   For any field $K$ and for any non zero polynomial $f\in K[X]$, there exists a field $L$ containing $L$ such that $f$ is decomposed into linear factors in $L$.

To prove it we use the following lemma.

LEMMA 0.5   For any field $K$ and for any irreducible polynomial $f\in K[X]$ of degree $d>0$, we have the following.

1. $L=K[X]/(f(X))$ is a field.
2. Let $a$ be the class of $X$ in $L$. Then $a$ satisfies $f(a)=0$.

Then we have the following lemma.

LEMMA 0.6   Let $p$ be a prime number. Let $q=p^r$ be a power of $p$. Let $L$ be a field extension of $\mathbb{F}_p$ such that $X^q-X$ is decomposed into polynomials of degree $1$ in $L$. Then

1. $$L_1=\{x \in L; x^q=x\}$$
   is a subfield of $L$ containing $\mathbb{F}_p$.
2. $L_1$ has exactly $q$ elements.

Finally we have the following lemma.

LEMMA 0.7   Let $p$ be a prime number. Let $r$ be a positive integer. Let $q=p^r$. Then we have the following facts.

1. There exists a field which has exactly $q$ elements.
2. There exists an irreducible polynomial $f$ of degree $r$ over $\mathbb{F}_p$.
3. $X^q-X$ is divisible by the polynomial $f$ as above.
4. For any field $K$ which has exactly $q$-elements, there exists an element $a\in K$ such that $f(a)=0$.

In conclusion, we obtain:

THEOREM 0.8   For any power $q$ of $p$, there exists a field which has exactly $q$ elements. It is unique up to an isomorphism. (We denote it by $\mathbb{F}_q$.)

The relation between various $\mathbb{F}_q$'s is described in the following lemma.

LEMMA 0.9   There exists a homomorphism from $\mathbb{F}_q$ to $\mathbb{F}_{q'}$ if and only if $q'$ is a power of $q$.