Q:

dedekind cut for the square root

Can somebody give me the hints to solve it ?

What is the Dedekind cut $(A, B)$ for  $\sqrt 2$ ?
What is the Dedekind cut $(C, D)$ for 
$\sqrt 2 + \sqrt 3$ ?
In $\mathbb R[x]/(x^2 + 1)$, what is the value of $[x^3 + x^2 + x + 1]$? 
(Just simplify $x^3 + x^2 + x + 1$)
where $[p]$ means the equivalence class that has $p$ as an element;
$[p]$ is the set of polynomials $q$ in $\mathbb R[x]$ such that
$x^2 + 1$ divides $p - q$.
$\mathbb R[x]$ is the set of polynomials whose coefficients are all real numbers.

A:

Try $A=\{x\in\mathbb Q\mid x^2 <2\}\cup  \mathbb Q_{\le 0}$. Why is "$\mathbb Q_{\le 0}$" needed? And what must $B$ look like? Verify that the properties of Dedekind cut are fulfilled!
Try $C=\{z\in\mathbb Q\mid \exists x\in\mathbb Q\colon x<z, x^2<2, (z-x)^2<3\}\cup \mathbb Q_{\le 0}$, so $D=\ldots$ Verify that the properties of Dedekind cut are fulfilled! And why is this the cut for $\sqrt 2+\sqrt 3$?
There is not really a value of $[x^3+x^2+x+1]$, but it is possible to find a representative of minimal degree. Just perform polynomial division to obtani $x^3+x^2+x+1=(x^2+1)(x+1)+0$, i.e. $[x^3+x^2+x+1]=[0]$

