Which subfields of C are closed under addition, subtraction, multiplication and division, but fail to contain 1.

Printable View

- Jun 13th 2009, 07:15 AMpoornaFields
Which subfields of C are closed under addition, subtraction, multiplication and division, but fail to contain 1.

- Jun 13th 2009, 07:36 AMTheAbstractionist
All subfields of $\displaystyle \mathbb C$ must contain 1. Or do you mean subrings of $\displaystyle \mathbb C\,?$

The only subring of $\displaystyle \mathbb C$ closed under division and not containing 1 is the trivial subring $\displaystyle \{0\}.$ (Note that division is not permitted in this ring and so it is vacuously closed under division.) For any other subring $\displaystyle R,$ if $\displaystyle a$ is a nonzero element, then the fact that $\displaystyle R$ is closed under division means that $\displaystyle 1=\frac aa\in R.$ - Jun 14th 2009, 07:06 AMpoorna
oh sorry, i meant subsets!

- Jun 14th 2009, 08:36 AMTheAbstractionist
Well, vacuously speaking, the empty set $\displaystyle \O$ 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.