nLab Ostrowski's theorem

Statement

Theorem

(Alexander Ostrowski)

Any non-trivial absolute value on the rational numbers is equivalent either to the standard real absolute value, or to the p-adic absolute value.

Theorem

(Ostrowski)

Any complete archimedean valued field is, as an associative algebra and as a topological space, isomorphic to either the real numbers or the complex numbers.

Related entries

• absolute value, archimedean valued field

• p-adic numbers

• analytic geometry

References

• Frédéric Paugam, p. 23 Global analytic geometry and the functional equation (2010) (pdf)

Discussion in point-free topology:

• Ming Ng, Steven Vickers, A Point-Free Look at Ostrowski’s Theorem and Absolute Values [arXiv:2308.14758]