Characteristic of Field

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.

$\blacksquare$

Exercises

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.

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Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.17$: Theorem $30$