Algebraic structures in RSA Why do we need a field for RSA and what are the two operations in this field?
Why can't we have a ring or group for example? Because in a group you also have inverse elements.
RSA is not done over field. When using RSA, you rely on the fact that the adversary does not know the order of the multiplicative group, i.e. he does not known the value of Euler's totient function and he is unable to compute it. 

Over field, computing it is trivial.