nLab Euclidean field


This entry is about a notion in algebra and analysis. For the notion in quantum field theory see at Euclidean field theory.






A Euclidean field is an ordered field FF with a principal square root function sqrt:[0,)F\mathrm{sqrt}:[0, \infty) \to F which satisfies the functional equation sqrt(x 2)=|x|\mathrm{sqrt}(x^2) = \vert x \vert for all xFx \in F, where x 2x^2 is the square function and |x|\vert x \vert is the absolute value. Here [0,)[0,\infty) denotes the non-negative elements of FF, hence, either {0}P\{0\}\cup P, or (P) c(-P)^c, for P={xFx>0}P=\{x\in F \mid x \gt 0\} the subset of positive elements, or {xFx0}\{x\in F\mid x\geq 0\} (in constructive mathematics some care will need to be taken in how this is defined.)


  • Every real closed field is a Euclidean field.

  • The real numbers constructible as lengths (or their negatives) via straightedge and compass from rational numbers.

See also


See also:

Last revised on May 9, 2022 at 00:50:16. See the history of this page for a list of all contributions to it.