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In the book "Linear Algebra" by Georgi E. Shilov, Chapter I the exclusion of $0$ from the numbers considered is justified by a note stating:

Given two elements $N$ and $E$, say, we can construct a field by the rules $N+N=E, N+E=E, E+E=N, N\cdot N=N, N\cdot E=N, E\cdot E=E$. 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.

What leads to the assumption that $E+E=N$?

terran
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AlexanderCar
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    Looks like there is a typo and it should say $N+N=N$ rather than $N+N=E$. – Eric Wofsey Sep 06 '23 at 18:40
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    I think you're misinterpreting the passage. The point is that we can have $1+1=0$ in a field if we don't impose an additional requirement. Does the author define "natural field element"? – Karl Sep 06 '23 at 18:41
  • @Karl The author defines natural numbers in a field $K$ as the numbers $1, 1+1, 1+1+1,\dots$ where $1$ is the multiplicative identity of the field. "Natural field element" is a field element which is natural. – JMoravitz Sep 06 '23 at 18:49
  • @ AlexanderCar, Given the first several assumptions and the assumption that there are only the two distinct elements $N$ and $E$, we must have that $E+E$ equals something. It must be that since this being a field we should have had then that $(-E)+E+E=(-E)+E$ would imply that $E=0$, but the earlier bit about $N+E=E$ would have meant that $N=0$ and we said earlier that $N\neq E$. – JMoravitz Sep 06 '23 at 18:52
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    For what its worth, I do not see this is something to typically exclude... Fields of characteristic 2 are still useful. I do not see any actual justification for not allowing zero as a natural number here. The author just arbitrarily began with defining naturals as $1,1+1,1+1+1,\dots$ (as is his right) but there is no good justification for why he made that choice beyond that it was convenient for him... just as it is convenient for many of the rest of us to include zero in what we call the natural numbers for other considerations. – JMoravitz Sep 06 '23 at 18:55
  • I don't consider $0$ to be a natural number because it appears relatively late in the history of mathematics. If you asked a layperson how many elves they see or how many of some other object that doesn't exist they would probably look at you funny. Many of would say zero because we know how to use zero but many would also consider it to be a meaningless question. – John Douma Sep 06 '23 at 19:30
  • @JMoravitz Depends on the context. I don't think anyone is disputing that $\mathbb{Z}_2$ is a field. The author just doesn't want finite fields for the current purpose. – aschepler Sep 06 '23 at 19:44
  • @JohnDouma History isn't a very strong argument. In some mathematical disciplines, the community norm is that $0 \in \mathbb{N}$; in others the norm is $0 \notin \mathbb{N}$ (though it's always advisable to be clear when it matters). I recall seeing a good list of what disciplines and similar contexts use each, but can't find it right now. – aschepler Sep 06 '23 at 19:51
  • @JMoravitz It makes sense but if N and E are the only elements in the set where does 2 come from later? – AlexanderCar Sep 06 '23 at 19:56
  • @aschepler It may not be for you, but it is for me. I know that $0\in\mathbb N$ is the norm but in my formal math studies, only computer scientists considered zero to be a natural number. I believe the addition of zero as a natural number didn't come until the computer science age. – John Douma Sep 06 '23 at 19:59
  • @JohnDouma Mine is a sort of "when in Rome, do as the Romans" reasoning. I'll agree to disagree. – aschepler Sep 06 '23 at 20:03
  • @JohnDouma: I doubt that what you say about the influence of computer science is true. My understanding is that including $0$ as natural number become popular in pure mathematics at some point in the early or mid twentieth century - and was implicit in the work of much earlier mathematicians. It was routinely taught in schools in the UK in the 70s and 80s but fell out of favour in school education at some point after that. The rejection of $0$ as a natural number has then followed on into higher education. ... – Rob Arthan Sep 06 '23 at 20:50
  • ... Personally, I think that, as we have the excellent phrase "positive integer" for the non-zero natural numbers, it is terminologically uneconomical not to take $0$ to be a natural number. I don't think the argument about laypeople is relevant: if you are doing combinatorics or probability (like Pascal, say), then $0$ surely is a natural number, just as an impossible event is an event (if not, then you are cutting the nose off his triangle). Similarly, Newton's work on the binomial theorem involves $0$ exponents. – Rob Arthan Sep 06 '23 at 20:50
  • @AlexanderCar "Where does 2 come from?" It is the symbol corresponding to the number defined as $1+1$, or $S(1)$ if you prefer where $S$ is the successor function, for whatever context and whatever definition of $1$ is appropriate given that context. – JMoravitz Sep 07 '23 at 01:30

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The author is saying that we are in general allowed to define $E+E=N$, and this results in a structure that satisfies the definition of a field. The smallest positive integer $p$ such that $p\cdot 1=0$ is called the characteristic of the field, and in that case the "natural number" $p$ is equal to $0$.

For whatever reason, the author doesn't want this to happen. A field where no sum of $1$s is $0$ is said to be of characteristic $0$. Basically the author is only considering fields of characteristic $0$.

Matt Samuel
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  • Though you also said "the smallest positive integer", so we should conclude $p$ doesn't exist rather than $p=0$. "Characteristic zero" is just a term we use for convenience that doesn't actually match the definition of "characteristic". – aschepler Sep 06 '23 at 19:42
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    @aschepler That's true. But $0$ has the convenient property that $0\cdot 1$ is also $0$, so we can in general say that multiplying $1$ by the characteristic of the field gives $0$. – Matt Samuel Sep 06 '23 at 19:47