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.

1. Exercise
Let $$Char(K) = 3$$, where $$K$$ is a field $$(K, +, .)$$.

Is {$${a^9 | a \in K}$$} a subfield of $$K$$?

1. Solution
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