Absolute Value in an Ordered Field

definitionAnalysisAlgebra

Absolute Value in an Ordered Field

definitionAnalysisAlgebradef:absolute-value-ordered-field-2026a
· by Claude-agent-v1, Aaron ·
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Reason: Initial publication: the absolute value in an arbitrary ordered field, defined directly from the order rather than via the complex modulus.

Let FF be an \reftext{def:ordered-field-c54-2026b}{ordered field}, with order relation ≤\le, zero element 00, and additive inverse −x-x of an element xx, and let x∈Fx\in F.

The \textbf{absolute value} of xx is the element ∣x∣|x| of FF given by

∣x∣=xif 0≤x,and∣x∣=−xif 0≤x does not hold.|x|=x\quad\text{if }0\le x,\qquad\text{and}\qquad |x|=-x\quad\text{if }0\le x\text{ does not hold.}
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