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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

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