In a field, the additive multiple $n\cdot 1_F$ is the canonical natural $\iota (n)$: the additive power of the group-power definition and the canonical natural are the same function, both being the unique one given by the recursion $\iota (0)=0_F$, $\iota (\sigma (n))=\iota (n)+1_F$

Let $F$ be a field (Field), which is a ring by Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring. Two functions $\mathbb{N}\to F$ are in play:

• $\iota$, the canonical natural of The canonical natural $\iota (n)=n\cdot 1_F$ of a field, defined by $\iota (0)=0_F$ and $\iota (n+1)=\iota (n)+1_F$;
• $n\mapsto n\cdot 1_F$, the additive natural power of the element $1_F$ in the abelian group $(F,+,0_F)$, defined by Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$ read additively: $0\cdot 1_F=0_F$ and $\sigma (n)\cdot 1_F=n\cdot 1_F+1_F$.

These are the same function: $\iota (n)=n\cdot 1_F$ for every $n\in \mathbb{N}$. In particular the notation $n\cdot 1_F$ used by The canonical natural $\iota (n)=n\cdot 1_F$ of a field and the notation $n\cdot 1_F$ used by Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$ denote the same element of $F$, and no second notion is in play.

• This is the item the characteristic rests on. The characteristic of a ring: the least $n\ge 1$ with $n\cdot 1_R=0$ when one exists, and $0$ otherwise is stated for an arbitrary ring using the additive powers of Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$; for a field a reader may already know The canonical natural $\iota (n)=n\cdot 1_F$ of a field, and without the present lemma the page would carry two notations for one element and invite the reader to assume they agree. That assumption is exactly the defect this item removes, and it is removed by a proof rather than by a remark.

• Uniqueness, not a computation, is what does the work. Both functions are characterised by the same recursion, and The recursion theorem says a recursion of that shape has exactly one solution. An induction on $\mathbb{N}$ would prove the same thing directly; nothing is gained by writing it out, since the uniqueness clause of The recursion theorem is that induction.

• $\mathbb{N}$ contains $0$, and $\iota (0)=0_F$, not $1_F$. So $\iota$ is not the map $n\mapsto$ "$1_F$ added to itself $n$ times" for $n\ge 1$ only: the value at $0$ is a genuine value of the recursion. The canonical natural $\iota (n)=n\cdot 1_F$ of a field records the same point, and the published Canonical naturals are positive and strictly increasing states its own recursion from $n=1$, which agrees because $\iota (1)=\iota (0)+1_F=1_F$.