I don't think excluding such fields is necessary in the same sense as excluding rings with 0 = 1. Requiring that 1 + 1 + ... + 1 ≠ 0 selects the fields with characteristic 0. From Wikipedia:

"In a finite field there is necessarily an integer n such that 1 + 1 + ··· + 1 (n repeated terms) equals 0. It can be shown that the smallest such n must be a prime number, called the characteristic of the field. If a (necessarily infinite) field has the property that 1 + 1 + ··· + 1 is never zero, for any number of summands, such as in Q, for example, the characteristic is said to be zero."