## Random nonzero elements in a finite field

### Idea

Let $q = p^n$ for some prime $p$ and some integer $n \ge 1$,
and let $F_q$ be the field with $q$ elements, and $F_p$ the
field with $p$ elements.

Recall that $F_q$ is a vector space over the prime field $F_p$,
and that for any generator $z$ of $F_q$ as a field extension of $F_p$,
the family $(1, z, ..., z^{n-1})$ is a basis of $F_q$ as a vector space
over $F_p$.

One way to build a list of random nonzero elements in $F_q$ is to pick
coefficients $a_0$, ..., $a_{n-1}$ at random, all between $0$ and $p-1$,
but not all zero, and to take the element $\sum a_k z^k$ in $F_q$.

One way to pick such a collection of coefficients is to pick an integer
at random between $1$ and $q-1$ and to let $a_k$ be its $k$-th digit
in base $p$.

### Implementation

Choose $p$ and $n$, define $q = p^n$, and let $F$ be the finite field with $q$ elements.

```
sage: p = 5
sage: n = 3
sage: q = p^n
sage: F = GF(q)
```

Choose $m$, the number of random elements to pick.

```
sage: m = 10
```

Produce a list of length $m$ of random nonzero elements in $F$.

```
sage: L = [F(ZZ.random_element(1, q).digits(base=p)) for _ in range(m)]
```

### Explanation

`r = ZZ.random_element(1, q)`

picks a random integer between $1$ and $q - 1$ `d = r.digits(base=p)`

gives the list of its digits in base $p$ `u = F(d)`

turns this list into a field element, seen a polynomial in $z$
(where $z$ is the generator of $F_q$ as a field extension of $F_p$)
with coefficients given by the list.

### Note on speed

Defining the finite field takes on the order of 0.1 ms.

It is probably worth defining $F$ once and for all, and then picking random
elements in $F$, rather than calling `GF(25)`

inside the loop, which spends
time initializing the finite field at each iteration of the loop.

Of course if you're looping only ten times it doesn't matter much.

You don't actually need the semicolons at the end of the lines, by the way.