ガロア第一論文と乗数イデアル他関連資料スレ2

[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ2 (1002ﾚｽ)
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359: １３２人目の素数さん [] 2023/03/13(月) 21:12:59.40 ID:UeELXD7y

>>358

つづき

The finish
For a, b in V define B(a, b) = (?ab ? ba)/2. Because of the identity (a + b)2 ? a2 ? b2 = ab + ba, it follows that B(a, b) is real. Furthermore, since a2 <= 0, we have: B(a, a) > 0 for a ≠ 0. Thus B is a positive definite symmetric bilinear form, in other words, an inner product on V.

Let W be a subspace of V that generates D as an algebra and which is minimal with respect to this property. Let e1, ..., en be an orthonormal basis of W with respect to B. Then orthonormality implies that:

e_{i}^{2}=-1,\quad e_{i}e_{j}=-e_{j}e_{i}.
If n = 0, then D is isomorphic to R.

If n = 1, then D is generated by 1 and e1 subject to the relation e2
1 = ?1. Hence it is isomorphic to C.

If n = 2, it has been shown above that D is generated by 1, e1, e2 subject to the relations

e_{1}^{2}=e_{2}^{2}=-1,\quad e_{1}e_{2}=-e_{2}e_{1},\quad (e_{1}e_{2})(e_{1}e_{2})=-1.
These are precisely the relations for H.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1677671318/359

360: １３２人目の素数さん [] 2023/03/13(月) 21:13:18.99 ID:UeELXD7y

>>359
つづき

If n > 2, then D cannot be a division algebra. Assume that n > 2. Let u = e1e2en. It is easy to see that u2 = 1 (this only works if n > 2). If D were a division algebra, 0 = u2 ? 1 = (u ? 1)(u + 1) implies u = ±1, which in turn means: en = ?e1e2 and so e1, ..., en?1 generate D. This contradicts the minimality of W.

Remarks and related results
The fact that D is generated by e1, ..., en subject to the above relations means that D is the Clifford algebra of Rn. The last step shows that the only real Clifford algebras which are division algebras are Cｌ0, Cｌ1 and Cｌ2.
As a consequence, the only commutative division algebras are R and C. Also note that H is not a C-algebra. If it were, then the center of H has to contain C, but the center of H is R. Therefore, the only finite-dimensional division algebra over C is C itself.
This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are R, C, H, and the (non-associative) algebra O.
Pontryagin variant. If D is a connected, locally compact division ring, then D = R, C, or H.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1677671318/360

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