Field with one element in an elementary context

Question

I've been reading about the "Field with one element", and I don't understand why is it a problem to consider $\{0\}$ as a field where $1=0$. I know in general the axioms don't let $1=0$, but why not? If a "field" has $1=0$, then it is necessarily the trivial field $\{0\}$ because $0=0a=1a=a$, so it wouldn't affect any other known fields.

Every explanation I've found about the field with one element involves category theory and advanced algebraic geometry, but I don't find any problem at an elementary level.

The nonzero elements of a field should be a multiplicative group. — Angina Seng, Apr 14 '18 at 11:48
"The nonzero elements of a field should be a multiplicative group." Why is that a problem. Wouldn't that be vacuously true? — fleablood, Jul 14 '20 at 17:04
@fleablood the way groups are usually defined require the existence of an identity element. If the the group is empty, there's no identity. — Javi, Jul 14 '20 at 20:48

score 11 · Accepted Answer · answered Apr 14 '18 at 13:19

The idea of the "field with one element" is not because we're interested in fields, but because we're interested in arithmetic geometry — the merger between number theory and algebraic geometry. Maybe there are other motivations too.

In both number theory and in algebraic geometry, fields — and the mathematical methods used in working with fields — are used extensively to describe various objects and features of the subject.

The finite fields are of particular interest for this. For example, the primes $2, 3, 5, 7, \ldots$ of the integers correspond to the finite fields $\mathbf{F}_2, \mathbf{F}_3, \mathbf{F}_5, \mathbf{F}_7, \ldots $.

One of the major obstacles in arithmetic geometry is that there is a big gaping hole in the theory, one you might expect to have something described by a field. And the really intriguing thing is that when mathematicians explore this hole, they find a lot of the features they expected to see.

Various numerology¹ arguments suggest if there were a field describing this hole, it would be a finite field of one element. To be clear, the zero ring does not have the features that mathematicians need.

These arguments have led to a whimsical description of this field of study as trying to work with a hypothetical object called $\mathbf{F}_1$. It's maybe more accurate to say that $\mathbf{F}_1$ refers to a hypothetical instance of a yet unknown² generalization of the notion of field.

_{1: I mean this in the whimsical sense that mathematicians use — e.g. doing suspect (and often nonsensical when taken literally) calculations to get an idea where to look for something more rigorous}

_{2: I am not an expert so I don't know the state of the art here. But my knowledge is simply that there are various candidate constructions that have some nice features, but none yet reproduce everything mathematicians are looking for}

score 1 · Answer 2 · answered Apr 28 '21 at 22:31

In fact what you need is a appropriate category of schemes over F1. Most of the theories, starting from the one developed by Deitmar, begin by working with F1 algebras, i.e. the idea is like work with modules on F1, that is, pointed sets. In this way, by endowing these algebras with a commutative product, it follows that the initial object of this category is the monoid with elements 0 and 1, and then in most of the theories the best model that exists of this hypothetical field is the initial object of the category of pointed commutative monoids.

Field with one element in an elementary context

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