grandes-ecoles 2023 Q24

grandes-ecoles · France · x-ens-maths-a__mp Groups Algebra and Subalgebra Proofs

We fix $\beta \in \ker(T - \mathrm{id}) \setminus \{0\}$. a) Show that the map $x \mapsto \beta x$ sends $\ker(T - \mathrm{id})$ into $\ker(T + \mathrm{id})$. Deduce that $\beta^2 \in U$ and that $\ker(T - \mathrm{id}) = \beta U$. b) Show that $\beta^2 \in ]-\infty, 0[$. c) Prove Theorem B: An algebraic $\mathbb{R}$-algebra without zero divisors is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$.

We fix $\beta \in \ker(T - \mathrm{id}) \setminus \{0\}$.\\
a) Show that the map $x \mapsto \beta x$ sends $\ker(T - \mathrm{id})$ into $\ker(T + \mathrm{id})$. Deduce that $\beta^2 \in U$ and that $\ker(T - \mathrm{id}) = \beta U$.\\
b) Show that $\beta^2 \in ]-\infty, 0[$.\\
c) Prove Theorem B: An algebraic $\mathbb{R}$-algebra without zero divisors is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$.

Paper Questions

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