Ordered field/Contains Q/Operations and ordering/Exercise

Let ${}K$ denote an ordered field. We consider the mapping ${}\mathbb {Z} \rightarrow K$ constructed in exercise.

a) Show that this mapping is injective.

b) Show that this mapping can be extended to an injective mapping ${}\mathbb {Q} \rightarrow K$ such that the addition and multiplication in ${}\mathbb {Q}$ and in ${}K$ coincide, and such that the ordering of ${}\mathbb {Q}$ coincides with the ordering of ${}K$.

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