Complete Archimedean Valued Field is Real or Complex Numbers
Jump to navigation
Jump to search
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
![]() | This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. 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 {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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 $1.1$