Definition 3

Let $K$ be an ordered field. An ordered extension of $K$ is a pair $(E, u)$, where $E$ is an ordered field and $u$ is an increasing homomorphism from $K$ to $E$.

Let $K$ be a field, let $E$ be an ordered field and let $u : K \to E$ be a homomorphism. The relation

is a total order relation on $K$ which gives it an ordered field structure, said to be induced by that of $E$. If $K$ and $E$ are ordered fields, then a homomorphism $u : K \to E$ is increasing if and only if the ordered field structure of $K$ is induced by that of $E$. We will usually identify $K$ with its image in $E$ under $u$.