What differences are there between $\mathbb Z_p$ and $\mathbb F_p$?

Question

I read some books about finite fields, sometimes the author refers to the finite field $\mathbb{F}_p$ and sometimes to the finite cyclic group $\mathbb Z_p$. What is the difference between them?

Answer 1

Roughly speaking, they are the same set but with different emphasis.

If we talk about the finite field $\Bbb F_p$, where $p$ is a prime, this can be visualised as the integers $\{0,1,\ldots,p-1\}$ with the two operations of addition modulo $p$ and multiplication modulo $p$. You can check that the field axioms hold true in this case.

If we talk about the finite cyclic group $\Bbb Z_p$ then we are still visualising the numbers $\{0,1,\ldots,p-1\}$, but we are only working with one operation, which would be addition modulo $p$. You can check that the group axioms are satisfied, and moreover that the group is cyclic.

Notice however that this only works if $p$ is prime. A finite field with $n$ elements $\Bbb F_n$ exists if $n$ is a power of a prime, $n=p^\alpha$, but if $\alpha>1$ then this is not the same as $\Bbb Z_n$.

The finite cyclic group $\Bbb Z_n$ exists for all positive integers $n$.

Examples:

• The finite field $\Bbb F_{31}$ and the cyclic group $\Bbb Z_{31}$ consist of the same numbers: the only real difference is that in $\Bbb F_{31}$ we consider problems involving multiplication and in $\Bbb Z_{31}$ we don't.
• There is a finite field $\Bbb F_{32}$ and a cyclic group $\Bbb Z_{32}$, but they are not at all the same.
• There is a cyclic group $\Bbb Z_{33}$, but there is no field $\Bbb F_{33}$ because $33$ is not a power of a prime.