Context: I am new to this. I started my course yesterday.
I know that the operations $+$ and $\cdot$ are required to satisfy the field axioms. So how can $F=\{0\}$ be a field? Recall the additive and multiplicative identity: there exist 2 different elements $0$ and $1$ in $F$ such that $ a + 0 = a$ and $a \cdot 1 = a$
But there is only one element in $F$? Shouldn't that mean that this rule has not been satisfied and thus $F$ is not a field?
My teacher mentioned something about $0=1$ can someone explain that?
Update my lecture notes:
Under the usual axioms $\{0\}$ is not a field. E.g., if $F$ is s field and $0$ its additive neutral then $F\setminus\{0\}$ is a group under multiplication - but groups cannot be empty. Put differently, one often takes $0\ne 1$ as one of the axioms of a field.