# 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

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 $\Char \C = 0$.

Fake Proof suggests: The validity of the material on this page is questionable.In particular: $\Char \C = 0$ is not a result from the infiniteness of $\C$. See Field has Algebraic Closure and Algebraically Closed Field is Infinite.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Questionable}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

From that contradiction follows the result.

$\blacksquare$

This article is complete as far as it goes, but it could do with expansion.In particular: See Field has Characteristic of Zero iff exists Monomorphism from Rationals.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Example $24$