1. Facts about finite fields

1. $\mathbb{Z}/p\/\mathbb{Z}$ is a field (which we'll call $\mathbb{F}_ p$ when talking about it as a field).

2. Each finite field has a prime characteristic.

3. If the characteristic of a finite field $F$ is $p$, then
   • $\mathbb{F}_ p$ is contained in $F$ as a subfield;

   • $F$ is a vector space over $\mathbb{F}_ p$;

   • if the vector-space dimension of $F$ over $\mathbb{F}_ p$ is $n$, then $F$ has $p^n$ elements.

4. For each possible prime power $p^n$,
   • there is a finite field of size $p^n$;

   • there is only one field of size $p^n$, ``up to isomorphism''. (This means that any two fields of that size must be isomorphic.)

   • Our name for this field is $\mathbb{F}_ {p^n}$. (Some books call the field GF$(p^n)$ for ``Galois field''.)

   • We have discussed the specific examples
     • $\mathbb{F}_ p$ ( $= \mathbb{Z}/p\/\mathbb{Z}$ ) for each prime $p$.

     • $\mathbb{F}_ 4$

     • $\mathbb{F}_ 8$

     • $\mathbb{F}_ {2^8}$ (in connection with Rijndael)

     • $\mathbb{F}_ 9$

5. The nonzero elements of a finite field (in other words, the units) have a generator $\alpha$ (also called a ``primitive element''). Equivalent ways to describe $\alpha$:
   • The powers of $\alpha$ are all the nonzero elements;

   • the order of $\alpha$ is $p^n - 1$, if the field has $p^n$ elements;

   • $1, \alpha, \alpha^2,\dots, \alpha^{p^n - 2}$ are all the nonzero elements. (Why $p^n - 2$ and not $p^n - 1$?)

6. If the field has $p^n$ elements, the generator $\alpha$ is also a root of some irreducible polynomial of degree $n$ in $\mathbb{F}_ p[x]$.

7. The powers $1, \alpha, \alpha^2,\dots, \alpha^{n-1}$ (the first $n$ powers) can be used as a basis for $\mathbb{F}_ {p^n}$ over $\mathbb{F}_ p$.

   The next power $\alpha^n$ can be re-expressed as a linear combination of lower powers using the irreducible polynomial.

8. In $\mathbb{F}_ {p^k}$, the function $\phi$ given by $\phi(a) = a^p$ is an ``automorphism''--an isomorphism of the field with itself.