Characteristic of Field is Zero or Prime
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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 Finite Ring with No Zero Divisors, if $\Char F \ne 0$ then it is prime.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Theorem $30$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 89 \alpha$