Let $A$ and consider $A^X$, with $X$ a non-empty a set. The operations are point-wise addition and multiplication. I have proved that $A^X$ is a ring, that it is commutative iff A is commutative and that it is unital iff A is unital
Now I am stuck in this part where they ask to prove or disprove: Is $A^X$ a field iff A is a field ? Does one of the implication holds? What can be concluded about A?
So far: I know that if A is field, $A^X$ will not be a field because I can always define a function that has 0 in the codomain in at least one point and therefore because 0 doesn't have a multiplicative inverse the whole function will be not invertible. Then the reverse and the double implication are false.
what about the forward implication?
If $A^X$ is a field, what can I say about $ A$? is it necessarily a field? or maybe just a division ring or a unital ring?