Characteristic of Subfield of Complex Numbers is Zero

From ProofWiki
Jump to navigation Jump to search


The characteristic of any subfield of the field of complex numbers is $0$.


Aiming for a contradiction, suppose to the contrary.

Let $K$ be a subfield of $\C$ such that $\Char K = n$ where $n \in \N, n > 0$.

Then from Characteristic times Ring Element is Ring Zero:

$\forall a \in K: n \cdot a = 0$

But as $K$ is a subfield of $\C$ it follows that $K \subseteq \C$ which means:

$\exists a \in \C: n \cdot a = 0$

Thus, by definition of characteristic:

$0 < \Char \C \le n$

But $\C$ is infinite and so $\Char \C = 0$.

From that contradiction follows the result.