Characteristic of Subfield of Complex Numbers is Zero

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Theorem

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


Proof

Suppose to the contrary.

Let $K$ be a subfield of $\C$ such that $\operatorname{Char} \left({K}\right) = n$ where $n \in \N, n > 0$.

Then:

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

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:

$0 < \operatorname{Char} \left({\C}\right) \le n$

But $\C$ is infinite and so $\operatorname{Char} \left({\C}\right) = 0$.

From that contradiction follows the result.

$\blacksquare$


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