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.

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Definition

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.)

Examples

• 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

• square root

• real square root

• Euclidean space

References

See also:

• Wikipedia, Euclidean field