Why does rational trigonometry not work over a field of Characteristic 2?

Squaring behaves strangely in characteristic 2. One result if this weirdness is the identity $(x+y)^2 = x^2 + y^2$ -- squaring doesn't result in the 'mixed terms' it usually does. Among the things that this "breaks" is the theory of quadratic forms and bilinear forms.

When you have a symmetric bilinear function -- that is a function $B(x,y)$ satisfying

• $B(x+y,z) = B(x,z) + B(y,z)$
• $B(x,y) = B(y,x)$
• $B(rx,y) = r B(x,y)$ where $r$ is a scalar

then you can construct a "quadratic form" $Q(x) = B(x,x)$. Conversely, when you have a quadratic form $Q(x)$, you can construct a function $B'(x,y) = Q(x+y) - Q(x) + Q(y)$.

These constructions are almost inverses: you have an identity $B'(x,y) = 2 B(x,y)$. So in any setting where $2$ is invertible, one can seamlessly pass back and forth between the idea of a quadratic form and the idea of a bilinear form.

But in characteristic $2$, the connection breaks, since the identity becomes $B'(x,y) = 0$.

Vector geometry relies heavily on multi-linear algebra: linear forms, bilinear forms, determinants, and so forth.

Rational trigonometry, is meant to more directly mimic classic trigonometry and relies very much on squaring to keep things rational. Quadrance is a quadratic form that is normally the one associated with the dot product, but that connection is broken in characteristic 2. Spread is more complicated, but I believe its connection with the cross product is also broken.

Circles become straight lines in characteristic $2$. Rotations, the linear transformations that keep a circle in place, become translations. This would tend to make trigonometry very trivial or very subtle, and definitely very different, from what we are accustomed to.

The equation $x^2 + y^2 = 1$ is the same as $(x+y)^2 = 1$ when $2=0$, which is the same as $(x+y - 1)^2 = 0$. This equation defines the same set of points as the line $x+y=1$. Algebraically it is a "double line".

Now, where

several published sources, claim that rational trigonometry does not work in fields (whether finite or infinite) of characteristic 2 "for technical reasons."

here are a few things that are not problems.

1. The formulas for the basic quantities of Rational Trigonometry, spread and quadrance, do not require division by $2$.

2. The formulas can be written, still not using division by $2$, so as to use only the dot products between vectors. If geometry is defined as the structure invariant under dot-product preserving transformations of the coordinate plane, the spread and quadrance are meaningful geometric quantities in characteristic $2$ for the same reason that they are geometric quantities in Euclidean geometry.

3. The equation "Spread = constant" continues to define a pair of straight lines relative to a fixed line.

It looks like the problem in characteristic $2$ is not that trigonometry, rational or not, does not work, but that the nonlinear geometry of circles is missing.