Modular aritmetic and fields

Question

I'm studying the concept of field applied to modular aritmetic. Is it correct to say that, if the dimension is a prime number $p$ then field properties are satisfied for the integers $\bmod p$ ? And that, if the dimension is a power of a prime number $p$ (that is $p^{n}$), not all integers $\bmod p^{n}$ form a field, but only a set of $p^{n}$ integers ?

Answer 1

The integers modulo $p$ for a prime $p$ does form a field.

The integers modulo $p^n$ for $n>1$ doesn't form a field. And there isn't a subset of that which forms a field either.

There are fields with $p^n$ elements for any prime $p$ and natural number $n$. However, for $n>1$ this field does not coincide with the ring of integers modulo $p^n$. For instance, the additive group of such a field is $(\Bbb Z_p)^n$.