Which subfields of C are closed under addition, subtraction, multiplication and division, but fail to contain 1.
All subfields of must contain 1. Or do you mean subrings of
The only subring of closed under division and not containing 1 is the trivial subring (Note that division is not permitted in this ring and so it is vacuously closed under division.) For any other subring if is a nonzero element, then the fact that is closed under division means that
Well, vacuously speaking, the empty set would satisfy the given conditions. Presumably, you want a nonempty subset. Any nonempty subset of a field that is closed under addition, subtraction and multiplication is a subring, so what I said in my previous post applies.