Why is $\displaystyle \mathbb{Z}$ not a field?
A field requires that every nonzero element has an inverse. The only invertable element of the integers are plus or minus one.
For example $\displaystyle 2$ is not invertible in $\displaystyle \mathbb{Z}$
$\displaystyle 2x=1$ does not have any integer solutions as $\displaystyle \frac{1}{2} \notin \mathbb{Z}$
Oh, OK. I was looking and looking at the definition of a field and comparing it to the properties of the integers and I couldn't find the missing ingredient. So, this little property of the integers, namely that of not having a multiplicitive inverse, is the only condition of the definition of a field that is not satitsfied, correct?