Is $(\mathbb{Z}/1\mathbb{Z}, + , \cdot)$ a field? I know that $(\mathbb{Z}/n\mathbb{Z}, + , \cdot)$ is only a field if $n$ is a prime number. 
What happens in case $n=1$? Is it still not a field? Because I think that the issue (with the inverses) does not occur for $n=1$, does it?

Here is one of the field axioms as listed in Wikipedia:

Additive and multiplicative identity: there exist two different elements $0$ and $1$ in $F$ such that [for all $a\in F$] $a + 0 = a$ and $a · 1 = a$.

$\mathbb Z/\mathbb Z$ doesn't satisfy this axiom because it doesn't have two different elements.
The convention that fields (and integral domains more generally) have at least two elements, or equivalently, that $0\neq 1$, is analogous to the convention of defining a prime number to not be $1$, and defining a prime ideal to not be the entire ring.  It avoids having to frequently make exceptions for the trivial case. 
