- #1
mathmari
Gold Member
MHB
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Hey! ![Eek! :eek: :eek:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
In my notes there is the following example:
$$\mathbb{Q}(\sqrt{2}) \overset{\widetilde{\sigma}}{\longrightarrow}\mathbb{R}\\ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \mathbb{Q} \overset{\sigma=id : q \mapsto q}{\rightarrow}\mathbb{R}$$
$p(x)=Irr(\sqrt{2}, \mathbb{Q})=x^2-2 \in \mathbb{Q}[x]$
$p^{\sigma}=x^2-2 \in \mathbb{R}[x]$ has two different roots in $\mathbb{R}$ : $ \pm \sqrt{2}$
So there are two embeddings $\widetilde{\sigma} : \mathbb{Q} (\sqrt{2}) \rightarrow \mathbb{R}$ :
- $\widetilde{\sigma}(\sqrt{2})=\sqrt{2}$ so $\widetilde{\sigma} ( \xi)=\xi, \forall \xi \in \mathbb{Q}(\sqrt{2})$
- $\widetilde{\sigma}(\sqrt{2})=-\sqrt{2}$ so $\widetilde{\sigma}(q_o+q_1 \sqrt{2})=q_0-q_1\sqrt{2}$
Could you explain me howwe found these two embeddings?? (Wondering)
In my notes there is the following example:
$$\mathbb{Q}(\sqrt{2}) \overset{\widetilde{\sigma}}{\longrightarrow}\mathbb{R}\\ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \mathbb{Q} \overset{\sigma=id : q \mapsto q}{\rightarrow}\mathbb{R}$$
$p(x)=Irr(\sqrt{2}, \mathbb{Q})=x^2-2 \in \mathbb{Q}[x]$
$p^{\sigma}=x^2-2 \in \mathbb{R}[x]$ has two different roots in $\mathbb{R}$ : $ \pm \sqrt{2}$
So there are two embeddings $\widetilde{\sigma} : \mathbb{Q} (\sqrt{2}) \rightarrow \mathbb{R}$ :
- $\widetilde{\sigma}(\sqrt{2})=\sqrt{2}$ so $\widetilde{\sigma} ( \xi)=\xi, \forall \xi \in \mathbb{Q}(\sqrt{2})$
- $\widetilde{\sigma}(\sqrt{2})=-\sqrt{2}$ so $\widetilde{\sigma}(q_o+q_1 \sqrt{2})=q_0-q_1\sqrt{2}$
Could you explain me howwe found these two embeddings?? (Wondering)