The Canonical Map from the Natural Numbers to a Field

Let $K$ be a field with multiplicative identity $1$, let $\mathbb{N}$ be the set of natural numbers, and for $n\in\mathbb{N}$ let $[n]$ be the initial segment of $\mathbb{N}$ determined by $n$.

For $n\in\mathbb{N}$ let $u^{n}:[n]\to K$ be the map with $u^{n}_{k}=1$ for every $k\in[n]$. The canonical map of $K$ is the map $\iota_{K}:\mathbb{N}\to K$ given by

$$\iota_{K}(n)=\sum_{k=1}^{n}u^{n}_{k},$$

the finite sum in $K$ of the family $u^{n}$.