This entry is about a notion in algebra and analysis. For the notion in quantum field theory see at Euclidean field theory.
symmetric monoidal (∞,1)-category of spectra
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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A Euclidean field is an ordered field $F$ with a principal square root function $\mathrm{sqrt}:[0, \infty) \to F$ which satisfies the functional equation $\mathrm{sqrt}(x^2) = \vert x \vert$ for all $x \in F$, where $x^2$ is the square function and $\vert x \vert$ is the absolute value. Here $[0,\infty)$ denotes the non-negative elements of $F$, hence, either $\{0\}\cup P$, or $(-P)^c$, for $P=\{x\in F \mid x \gt 0\}$ the subset of positive elements, or $\{x\in F\mid x\geq 0\}$ (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:
Last revised on May 9, 2022 at 00:50:16. See the history of this page for a list of all contributions to it.