Characteristic of Field by Annihilator

Theorem

Let $\struct {F, +, \times}$ be a field.

Then of the following two cases, exactly one applies:

Suppose that:

$\map {\mathrm {Ann} } F = \set 0$

That is, the annihilator of $F$ consists of the zero only.

Then:

$\Char F = 0$

That is, the characteristic of $F$ is zero.

Suppose that:

$\exists n \in \map {\mathrm {Ann} } F: n \ne 0$

That is, there exists (at least one) non-zero integer in the annihilator of $F$.

If this is the case, then the characteristic of $F$ is non-zero:

$\Char F = p \ne 0$

and the annihilator of $F$ consists of the set of integer multiples of $p$:

$\map {\mathrm {Ann} } F = p \Z$

where $p$ is a prime number.

1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties