Classifying Algebraic Structures as Fields I just have a quick question!
We got a question asking us which of the following algebraic structures were fields, I got every one of them except the last one.  I don't even know how to begin how to attack the question.
Is real numbers modulo 2$\pi$ a field?

If you want the quotient to be a ring, then you must take the quotient by an ideal. The ideal generated by $2\pi$ is all of $\Bbb R$, so this quotient is the zero ring, which is not considered to be a field.
This is nothing special with $2\pi$: every nonzero number will make the quotient zero. If you try to mod out by zero, you will of course get just $\Bbb R$ back, and so that would be a field!
If you have convinced yourself the thing you are modding out is not all of $\Bbb R$, then unfortunately you are thinking of something which isn't an ideal, and so the quotient by this object will not even be a ring (and of course could not be a field, then.)
