[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)

ガロア第一論文と乗数イデアル他関連資料スレ2 http://rio2016.5ch.net/test/read.cgi/math/1677671318/

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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

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