# Diagrams characterizing ring characteristic, and in particular field characteristic 1?

1. For a given prime $p$, what is the diagram for saying (e.g. in some category where morphisms are field homomorphisms) that a field has characteristic $p$?
2. Is there a diagram, or the shape of what such a diagram would look like, for $F_{un}$, the field of characteristic 1? More directly, is $F_{un}$ characterized by a diagram?
• I don't understand what you want. Oct 23, 2014 at 3:31
• Moreover, the field with one element is not a field (and it may not be at all, for all we know...) Oct 23, 2014 at 3:33
• 1. If $K$ is a field with characteristic $p$, then $\mathcal i:\mathbb Z/p\mathbb Z\to K$ given by $\mathcal i(\overline n)=n\cdot1_K$ is a such diagram. It is clear that this approach is completely useless to solve your second question. Oct 23, 2014 at 3:33
• @MarianoSuárez-Alvarez: I'm asking for a way of looking at the field of characteristic 1 in terms of morphisms. But I guess that's silly, because it doesn't have morphisms to or from anything else? Oct 23, 2014 at 5:32
• The notion of $F_{un}$ is not vague. Many frameworks have been built up (see the paper "Mapping F1-land" for example) in which precise and rigorous definitions of $F_{un}$ are given. Although Mariano is right that $F_{un}$ is not a field, it is a "generalized" field and in each proposed framework $F_{un}$ shares properties which fields in the usual sense have; for example, every module over $F_{un}$ should be free. Oct 23, 2014 at 8:26