[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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358(3): 2023/03/13(月)21:11 ID:UeELXD7y(5/14) AAS
>>357
つづき

The claim
The key to the argument is the following

Claim. The set V of all elements a of D such that a2 <= 0 is a vector subspace of D of dimension n - 1. Moreover D = R 〇+ V as R-vector spaces, which implies that V generates D as an algebra.
Proof of Claim: Let m be the dimension of D as an R-vector space, and pick a in D with characteristic polynomial p(x). By the fundamental theorem of algebra, we can write

p(x)=(x-t_{1})\cdots (x-t_{r})(x-z_{1})(x-{\overline {z_{1}}})\cdots (x-z_{s})(x-{\overline {z_{s}}}),\qquad t_{i}\in \mathbf {R} ,\quad z_{j}\in \mathbf {C} \backslash \mathbf {R} .
省9

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359(3): 2023/03/13(月)21:12 ID:UeELXD7y(6/14) AAS
>>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.
省6

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368: 2023/03/14(火)07:40 ID:bQV51cAg(1/8) AAS
>>367
１、検索結果を読んでも全く理解できず全コピペ

さすが大学１年の４月で落ちこぼれた真正●●

Q,　>>357-360を読んで肝心な部分をまとめて
　　　2048バイト以内（すなわち１コメント）で書け

１には絶対できないと予言する

勝った！（完全勝利宣言！！！）

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381(8): 2023/03/14(火)20:13 ID:bQV51cAg(3/8) AAS
>>377-379
東京●●大と大阪●●大
落ちこぼれ同士の共鳴

> 数学科以外で自分より上がいると、
> 落ちこぼれた自分がみじめで許せないんだ
　誰が上？貴様が？
　正則行列も知らず
省19

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