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I'm wondering if there is existing terminology to describe fields $F$ with the properties below. I don't have a completely precise description of the concept I have in mind, but hopefully this will be ...

There are two integers: $A, B$. Given the below four allowed operations (and only them): $A+1$, $A-1$, $\sqrt{A}$, $A^2$ Also, it is only allowed to take the square root of $A$ when this square root ...