Characteristic of Field is Zero or Prime

Theorem
Let $F$ be a field.

Then the characteristic of $F$ is either zero or a prime number.

Proof
From the definition, a field is a ring with no zero divisors.

So by Characteristic of Ring with No Zero Divisors, if $\operatorname{Char} \left({F}\right) \ne 0$ then it is prime.

Exercise 1.
Let $\operatorname{Char} \left({K}\right) = 3$, where $K$ is a field $\left({K, +, \times}\right)$.

Is $\left\{{a^9 : a \in K}\right\}$ a subfield of $K$?

Solution 1.
?