Proof-method: Show that if $n$ is not prime then $\Bbb Z /n\Bbb Z$ is not a field The question pertains to the question in the title. That is, > Show that if $n$ is not prime then $\Bbb Z /n\Bbb Z$ is not a field. I realise that this has many duplicates, but my question is rather this; > Why does it not suffice to say that if $n$ is not prime then $\lvert\Bbb Z /n\Bbb Z \rvert = n \neq p^m$ for some prime $p$ and integer $m$ and so is not a field by definition? I realise this somewhat misses the _reason_ why it's not a field, but would it suffice as an answer to this question?

A major reason why you cannot use this argument is that you seem to saying that because you know every finite field must have $p^{m}$ elements you get a contradiction.

However how do you know that you don't need this result in order to prove that finite field must have $p^{m}$ elements.

So does the proof of finite field must have $p^{m}$ elements, require the knowledge of the prime subfield and does this in turn require knowing that $\mathbb{Z}_{n}$ is a field iff $n$ is prime...