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 $\mathbb{R}|\mathbb{Q}$

#Algebra #Analysis

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$