# What is the first order logic statement of “this field is of characteristic zero”?

I want to state that a field $$F$$ is of characteristic zero in logical notation to an audience without referring them to the meaning of the characteristic of a field.

My first thought was the proposition $$\forall x \in F \setminus \{-1\} : x + 1 \ne 0,$$ but this holds for any field.

How can “of characteristic zero” be stated using quantifiers?

• An equivalent statement is $F$ contains a subfield isomorphic to $\mathbb Q$ – Bang Pham Khoa Jun 4 at 2:09

## 2 Answers

No. In any field the equation $$x + 1 = 0$$ has the unique solution $$x = -1$$, independent of the characteristic.

If your audience is unfamiliar with the characteristic of a field and you need to explain it, do that with an example. In $$\mathbb{Z}_5$$ you have $$1 + 1 + 1 +1 + 1 = 0$$. In a field with characteristic $$0$$ no sum of $$1$$s can be $$0$$.

Don't use logical symbols, use words.

Edit in response to the edited question.

If you must use a formal statement you might say

For every positive integer $$n$$ the sum of $$n$$ $$1$$s is not $$0$$.

or

The smallest subfield is infinite.

• It's going in a list of axioms all written in the form $\forall a, b, c, \ldots \in F : P(a, b, c, \ldots)$. Would it suffice to write $\forall n \in \mathbb{N} : \sum_{i=1}^n 1 \ne 0$? – holomenicus Jun 4 at 2:42
• @holomenicus That should do, with the implicit assumption that $0 \not \in \mathbb{N}$. I'm not a logician - if first order logic is what you need I trust that sum as a function of $n$ can be written in first order logic. – Ethan Bolker Jun 4 at 3:01

No.

The statement is equivalent to

for all $$x\in F$$, if $$x+1=0$$ then $$x=-1$$.

This is true in every field.

To explain characteristic zero, how about just "adding $$1$$ and $$1$$ and $$1$$ and so on [finitely many times] will never give you zero". Omit the "finitely many times" if you think it will confuse your audience. And I would certainly suggest briefly giving an example of a field with non-zero characteristic, otherwise they will just say "that's obvious, what's the point?"

• +1 I was just finishing the same answer when yours appeared. – Ethan Bolker Jun 4 at 2:12