Complete Archimedean Valued Field is Real or Complex Numbers

Theorem

Let $\struct{k, \norm{\,\cdot\,}}$ be a complete valued field with Archimedean norm $\norm{\,\cdot\,}$.

Then either:

$k$ is isomorphic to the real numbers $\R$ and $\norm{\,\cdot\,}$ is equivalent to the absolute value $\size{\,\cdot\,}$ on $\R$

or

$k$ is isomorphic to the complex numbers $\C$ and $\norm{\,\cdot\,}$ is equivalent to the complex modulus $\size{\,\cdot\,}$ on $\C$

Proof

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Also see

This result is sometimes called Ostrowski's theorem.

In the same paper, published in $1918$, Ostrowski also proved that every non-trivial norm on the rational numbers is equivalent to either the $p$-adic Norm for some prime $p$ or the absolute value.

It is this latter result that is more often referred to as Ostrowski's Theorem.

Sources

• 1986: John William Scott Cassels: Local Fields: $\S 1$ Introduction, Theorem $1.1$