22 One square and an odd number of triangles
TODO
TODO
For any blue point \(p_0 = (x_b, y_b)\), green point \((x_g, y_g)\), and red point \((x_r, y_r)\), the \(v\)-value of the determinant
is at least \(1\).
TODO
Any line of the plane receives at most two different colors. The area of a rainbow triangle cannot be \(0\), and it cannot be \(\frac{1}{n}\) for odd \(n\).
Follow from 22.4
Every dissection of the unit square \(S = [0, 1]^2\) into finitely many triangles contains an odd number of rainbow triangles, and thus at least one.
TODO
It is not possible to dissect a square into an odd number of triangles of equal algebra area.
TODO
Appendix: Extending valuations
A proper subring \(R\subset K\) is a valuation ring with respect to some valuation \(v\) into some ordered group \(G\) if and only if \(K = R \cup R^{-1}\).
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The field of real numbers \(\mathbb {R}\) has a non-Archimedean valuation to an ordered abelian group
such that \(v(\frac{1}{2}) {\gt} 1\).
TODO