Field with one element in an elementary context 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. 

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.
