Properties of Natural Number Powers in a Field

Let $K$ be a field, with additive identity $0$ and multiplicative identity $1$, and let $c,d\in K$. Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, and let $n\in\mathbb{N}$. Powers are those of Natural Number Power of an Element of a Field.

Then the following hold.

1. (Recursion)¶ $c^{1}=c$, and $c^{S(n)}=c^{n}\,c$.

2. (Unit)¶ $1^{n}=1$.

3. (Products)¶ $(c\,d)^{n}=c^{n}\,d^{n}$.

4. (Vanishing)¶ $c^{n}=0$ if and only if $c=0$.

5. (Nonnegative bases)¶ Suppose $K$ is the field of real numbers, with the order $\le$ of its ordered field structure, and that $0\le c$. Then $0\le c^{n}$; and if in addition $c\le d$, then $c^{n}\le d^{n}$.

6. (Exponents)¶ $c^{m+n}=c^{m}\,c^{n}$ for every $m\in\mathbb{N}$, the sum $m+n$ being that of the natural numbers.

7. (Monotonicity in the exponent)¶ Suppose $K$ is the field of real numbers, with the order $\le$ of its ordered field structure, and that $1\le c$. Then $c^{m}\le c^{n}$ for every $m\in\mathbb{N}$ with $m\le n$, where $\le$ on $\mathbb{N}$ is the order on the natural numbers.