you wrote "the field with one element F1 does not exist.", however just now @MartinBrandenburg on this question[1] wrote to me "Of course F1 exists, in many ways." After looking into things myself, it appears to me that some authors dont make the rule $1 \neq 0$ in the field axioms, but then every theorem/proof needs to make a remark about the zero ring being excluded. Therefore I am leaning towards F1 does exist as a logical concept
[1] https://math.stackexchange.com/questions/4983752/what-is-the-reason-for-needing-to-have-an-element-1-neq-0-in-the-commutative?noredirect=1#comment1067678…
[1] https://math.stackexchange.com/questions/4983752/what-is-the-reason-for-needing-to-have-an-element-1-neq-0-in-the-commutative?noredirect=1#comment1067678…