Why must natural field elements be nonzero?

Hello, I'm reading Georgi Shilov's *Linear Algebra *and it states:

"The numbers 1, 1+1=2, 2+1=3, etc. are said to be *natural*; it is assumed that none of these numbers is zero. Given two elements N and E, say, we can construct a field by the rules N+N=N, N+E=E, E+E=N, N*N=N, N*E=N, E*E=E. Then, in keeping with our notation, we should write N=0, E=1 and hence 2=1+1=0. To exclude such number systems, we require that all natural field elements be nonzero."

I am wondering why this is necessary. (I'm new to abstract algebra.) Why do we want to exclude such number systems? Is it just for convenience or is it because they actually create a contradiction?

Re: Why must natural field elements be nonzero?

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."

Re: Why must natural field elements be nonzero?

Thank you so much! That clears it up. I do have a related question now, though. The book then goes on to say that, given that none of the natural numbers is zero, "it follows that every field K has a subset (subfield) isomorphic to the field of rational numbers." If I understand this correctly it means that the rationals are the "smallest" field of characteristic zero. But how does this follow?

Re: Why must natural field elements be nonzero?

It helps to think about what seems "natural". It seems natural to count objects which can be seen. It does not make sense to count something that isn't there, it is not natural to do so.

Re: Why must natural field elements be nonzero?

Quote:

Originally Posted by

**Ragnarok** The book then goes on to say that, given that none of the natural numbers is zero, "it follows that every field K has a subset (subfield) isomorphic to the field of rational numbers." If I understand this correctly it means that the rationals are the "smallest" field of characteristic zero. But how does this follow?

Yes, "[t]he rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of **Q**" (Wikipedia). (You can guess which of my skills is stronger: knowledge of abstract algebra or web-searching. :-)) Indeed, every field contains -1 as the additive inverse to 1. Multiplying -1 by natural numbers gives us integers, and dividing integers by natural numbers gives us rational numbers.