MHB How Do We Find the Embeddings of $\mathbb{Q}(\sqrt{2})$ in $\mathbb{R}$?

  • Thread starter Thread starter mathmari
  • Start date Start date
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

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)
 
Physics news on Phys.org
Hi,

Those morphisms sends roots of the polynomial of the primitive element into roots of the "image of the polynomial" (the polynomial with coefficients the image of the initial coefficients).

And the morphisms are defined by the image of the elements of a base (In this case, $\{1, \sqrt{2}\}$), as you know, these morphism needs to be the identity when restricted over the base field, so $\tilde{\sigma}(1)=1$ and you got two options to choose the image of $\sqrt{2}$ that are the two roots of $x^{2}-2$, i.e. $\tilde{\sigma}(\sqrt{2})=\pm \sqrt{2}$
 
Thread 'Derivation of equations of stress tensor transformation'
Hello ! I derived equations of stress tensor 2D transformation. Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture. I want to obtain expression that connects tensor for case 1 and tensor for case 2. My attempt: Are these equations correct? Is there more easier expression for stress tensor...
Back
Top