Non-linear cyclic order of a field

Question

A cyclic order of a group is non-linear if any cut of it is not compatible with the group operation.

A cut of a cyclically ordered set is a linear order $<$ such that

• $a < b < c \implies [a, b, c]$

for any elements $a$, $b$, $c$ of the set.

A cut of a cyclically ordered group is compatible with the group operation ($+$) iff:

• $a < b \implies a + x < b + x$ and $x + a < x + b$

for any elements $a$, $b$, $x$ of the group.

I am trying to find properties of a non-linear cyclic order on fields:

1. Is there an infinite field with a non-linear cyclic order on the additive group?
2. Is there a field with a non-linear cyclic order on the additive group such that $[0, 1, -1]$ and $[0, x, 1]$ for some element $x$?

Answer 1

By picking any irrational real number $\alpha$, there is an injective map

$$ \mathbb{Q} \to \mathbb{T} : x \mapsto e^{2 \pi i \alpha x} $$

and thus (?) $\mathbb{Q}$ inherits a cyclic ordering from the one on $\mathbb{T}$. Since, in the comments, you say you only care about compatibility with addition, I imagine this serves as an example to both of your questions.