Prove that there is a non-archimedean euclidean field K' containing R (t)

Question

In page 161 of Euclid and Beyond(robin hartshorne),

There is a problem saying that 'Prove that there is a non-archimedean euclidean field K' containing $\mathbb{R}(t)$.'

Here, t is any real number. The author defines $C$ as the set of continuous real-valued functions defined on some interval $(a_0, \infty )$ of $\mathbb {R}$ that are never 0. Then He defines K' as the set of all elements of C that can be obtained from $\mathbb {R} (t)$ by a finite mumber of operations +, -, ×, ÷, and a $\in$ $C$ with $a>0$ implies $\sqrt a$ $\in C$

I want to find the additive identity to prove that K' is a field, but I don't have any idea..

Answer 1

I want to find the additive identity to prove that K' is a field

The additive identity is clearly the constant zero function, which is equal to $f-f$ for any $f$ in your generating set. ($f-f$ is clearly a finite combination of things in $C$ with those operations.)