Fields

• Jun 13th 2009, 08:15 AM
poorna
Fields
Which subfields of C are closed under addition, subtraction, multiplication and division, but fail to contain 1.
• Jun 13th 2009, 08:36 AM
TheAbstractionist
All subfields of $\mathbb C$ must contain 1. Or do you mean subrings of $\mathbb C\,?$

The only subring of $\mathbb C$ closed under division and not containing 1 is the trivial subring $\{0\}.$ (Note that division is not permitted in this ring and so it is vacuously closed under division.) For any other subring $R,$ if $a$ is a nonzero element, then the fact that $R$ is closed under division means that $1=\frac aa\in R.$
• Jun 14th 2009, 08:06 AM
poorna
oh sorry, i meant subsets!
• Jun 14th 2009, 09:36 AM
TheAbstractionist
Well, vacuously speaking, the empty set $\O$ 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.