Field/N and Z/Canonical mapping/Exercise

Let ${}K$ be a field. Show that, for every natural number ${}n\in \mathbb {N}$, there exists a field element ${}n_{K}$ such that ${}0_{K}$ is the null element in ${}K$, and ${}1_{K}$ is the unit element in ${}K$, and such that

${}(n+1)_{K}=n_{K}+1_{K}\,$

holds. Show that this assignment has the properties

$(n+m)_{K}=n_{K}+m_{K}{\text{ and }}(nm)_{K}=n_{K}\cdot m_{K}.$

Extend this assignment to all integer numbers ${}\mathbb {Z}$, and show that the stated structural properties hold again.

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