I'm reading the Dover linear algebra book by Georgi E Shilov. In Chapter 1 the book discusses number fields. Regarding the assumption that natural numbers are non-zero, it says:
Given two elements $N$ and $E$, say, we can construct a field by the rules $N+N = N$, $N+E=E$, $E+E=N$, $N\cdot N=N$, $N\cdot E=N$, $E\cdot E=E$. 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 understand that the implication $2=0$ is undesirable. However, since the set of natural numbers is not defined by said rules, I don't understand what the pertinence is here. That is to say, why can't the same thing be said about integers and real numbers, etc? What am I missing?
When he says “The numbers $1$, $1+1=2$, $2+1=3$, etc. are said to be natural; it is assumed that none of these numbers are zero”, the symbol “$1$” and the word “zero” refer to the multiplicative and additive identity elements in the field $K$ in question (and the symbols “$2$” and “$3$” are defined to mean $1+1$ and $1+1+1$, respectively, with the addition in the field $K$).
As the example in the footnote shows, there are fields where $1+1=0$, and what he's saying is that he will only consider fields $K$ where such things do not happen, i.e., fields where $1+\dots+1 \neq 0$ no matter how many ones you add up. (Fields of characteristic zero, as they are called.)