Though many of the other answers are correct at the semantic level, I find it interesting to also approach these sorts of questions from the implementation details level.
An interface can be thought of as a collection of slots, which contain methods. When a class implements an interface, the class is required to tell the runtime how to fill in all the required slots. When you say
interface IFoo { void M(); }
class Foo : IFoo { public void M() { ... } }
the class says "when you create an instance of me, stuff a reference to Foo.M in the slot for IFoo.M.
Then when you do a call:
IFoo ifoo = new Foo();
ifoo.M();
the compiler generates code that says "ask the object what method is in the slot for IFoo.M, and call that method.
If an interface is a collection of slots that contain methods, then some of those slots can also contain the get and set methods of a property, the get and set methods of an indexer, and the add and remove methods of an event. But a field is not a method. There's no "slot" associated with a field that you can then "fill in" with a reference to the field location. And therefore, interfaces can define methods, properties, indexers and events, but not fields.
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