Why can't class fields be var?

Eric Lippert answered your question right here: Why no var on fields?

Basically, for the general case it would require re-writing the C# compiler, as the way it currently does type inference would not work for cycles of var field variable assignments.

The var keyword was designed for anonymous types, which can only be used inside of a method.

Also, you're wrong; the compiler cannot always deduce a var field.

What happens if you compile the following:

class A {
    public readonly var value = B.value;
}

class B {
    public readonly var value = B.value;
}

This situation is impossible to recreate with local variables, since a variable cannot be referenced before it's defined.

The general problem here is that you're asking the compiler to consume type information while it's still generating that information.
Eric Lippert explains in greater depth.

I see two reasons:

1. It might be desirable to make the declaration of types in a public interface explicit
2. It's hard to implement. The C# compiler compiles in multiple phases.
   At first it parses everything apart from method bodies so it knows about everything outside of function bodies. Then it can use that information to compile method bodies individually. What happens while compiling one method body hasn't much effect on what happens when compiling other method bodies.
   If you could use var for fields the expression body of the field initializer would affect the type of the field and thus many other methods. So it doesn't fit the compiler design well.