Proof-method: Show that if $n$ is not prime then $\Bbb Z /n\Bbb Z$ is not a field

Question

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?

Answer 1

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...

Answer 2

Because this is not enough to say that $\mathbb Z/9\mathbb Z $ is not a field, as $9$ is in fact of the form $p^m$ for a prime $p$ and positive integer $m$.