×

# A Real Injection

Suppose that $$(K,\pi,\sigma,1,0,\geq)$$ is an ordered field. My question is does there exist a complete ordered field $$(K',\pi',\sigma',1',0',\geq')$$ and an injective map $$\phi :K \longrightarrow K'$$ such that $$\phi(\pi(a,b)) = \pi'(\phi(a),\phi(b)), \phi(\sigma(a,b)) = \sigma'(\phi(a),\phi(b)), a \geq b \Longrightarrow \phi(a) \geq' \phi(b)$$ Because if there is then since ever ordered field is of characteristic zero (because $$0<1<2<... \Longleftarrow [1 \neq 0 \Longleftarrow 1 = 1^{2} > 0]$$ then $$(K,\pi,\sigma,1,0,\geq)$$ would be isomorphic to one of the intermediate fields of $$\mathbb{R}|\mathbb{Q}$$

Note by Samuel Queen
2 years, 10 months ago