23/03/13 21:11:19.15 UeELXD7y.net
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（参考）英語版に詳しい証明がある、ただし文字化けなおさず。本文参照ください
URLﾘﾝｸ(en.wikipedia.org)(real_division_algebras)
Frobenius theorem (real division algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following:
R (the real numbers)
C (the complex numbers)
H (the quaternions).
These algebras have real dimension 1, 2, and 4, respectively. Of these three algebras, R and C are commutative, but H is not.
Proof
The main ingredients for the following proof are the Cayley?Hamilton theorem and the fundamental theorem of algebra.
Introducing some notation
Let D be the division algebra in question.
Let n be the dimension of D.
We identify the real multiples of 1 with R.
When we write a <= 0 for an element a of D, we tacitly assume that a is contained in R.
We can consider D as a finite-dimensional R-vector space. Any element d of D defines an endomorphism of D by left-multiplication, we identify d with that endomorphism. Therefore, we can speak about the trace of d, and its characteristic and minimal polynomials.
For any z in C define the following real quadratic polynomial:
Q(z;x)=x^{2}-2\operatorname {Re} (z)x+|z|^{2}=(x-z)(x-{\overline {z}})\in \mathbf {R} [x].
Note that if z ∈ C ? R then Q(z; x) is irreducible over R.
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