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The problem is that vector spaces over char 2 base fields admit no reflections, so the theorem fails for boring reasons. Here's one way of seeing this: take F to be characteristic 2 and let V = F^n. If f is a proper reflection then there exists some basis in which it acts as (x_1, ..., x_k, x_(k+1)
200:(-x_1, ..., -x_k, x_(k+1), ..., x_n) where k is odd. But x = -x in characteristic 2, so f is the identity, not a reflection--contradiction! A similar proof works for any vector space over a char 2 base field.
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I can confirm that. Artin in
Geometric Algebra (New York 1957) specifically says it is due to E.Cartan and J.Dieudonné.13:42, 19 April 2007 (UTC)
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Does anyone have a reference that discusses the unique exception to this thereom in characteristic 2? --
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Knowledge. If you would like to participate, please visit the project page, where you can join
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but I am not perfectly sure. Can anyone confirm this ?
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