Characteristic of Subfield of Complex Numbers is Zero

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 $24$