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If $\mathbb{K}$ is any field and is endowed with the discrete topology, then $\mathbb{K}$ is a local field (*). further, if $\mathbb{K}$ is connected, then $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$ (**).

I was able to prove (*), but not (**).

edited: I am reading an article and above sentences are of the following paragraph of the article: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a local field if both $\mathbb{K}^+$ and $\mathbb{K}^*$ are locally compact Abelian groups, where $\mathbb{K}^+$ and $\mathbb{K}^*$ denote the additive and multiplicative groups of $\mathbb{K}$, respectively. If $\mathbb{K}$ is any field and is endowed with the discrete topology, then $\mathbb{K}$ is a local field. Further, if $\mathbb{K}$ is connected, then $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$. If $\mathbb{K}$ is not connected, then it is totally disconnected. Hence by a local field, we mean a field $\mathbb{K}$ which is locally compact, non-discrete and totally disconnected. The p-adic fields are examples of local fields.

Ehsan zarei
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    Um, what? Something with the discrete topology can't be connected if it has more than one point (which a field necessarily have). So it is vacuously true that any field that is connected in the discrete topology is either $\mathbb R$ or $\mathbb C$. (It's also made of green cheese). – hmakholm left over Monica May 23 '15 at 14:41
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    As far as I know, a local field has to be a locally compact one wrt a non-discrete topology... – Timbuc May 23 '15 at 14:45
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    In the second statement, you're missing the hypothesis that the topology on $K$ is locally compact (http://www.encyclopediaofmath.org/index.php/Topological_field); otherwise there are counterexamples. I don't understand the first statement at all. – Qiaochu Yuan May 23 '15 at 17:44

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For (*), I'm surprised by the definition you cite: a local field is usually defined as "a topological field with respect to a locally compact nondiscrete topology (with some extra properties)" (http://en.wikipedia.org/wiki/Local_field). If we don't require indiscretion, then your statement is trivially true since discrete topologies are locally compact; otherwise your statement is even more trivially true since there are no discrete local fields at all.

I'm interpreting (**) as being, "If $K$ is any connected local field, then $K$ is either $\mathbb{R}$ or $\mathbb{C}$." As Qiaochu points out, this statement requires that local compactness be part of the definition of local field (that is, it's false for merely topological fields). Under this assumption, the statement is true; see Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?.

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Noah Schweber
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  • Actually, this question might be a duplicate of http://math.stackexchange.com/questions/334078/are-bbb-r-and-bbb-c-the-only-connected-locally-compact-fields? – Noah Schweber May 23 '15 at 19:58