Quadratic Fields in the "Revista Matematică (a Elevilor) din Timişoara

           Quadratic fiels are a very serious mathematical notion. A book accessible to any schoolchild is Mak TRIFKOVIĆ - Algebraic Theory of Quadratic Numbers (Springer Science+Business Media, New York, 2003).

           RMT is a magazine.. not too serious: don't try to use the electronic edition because you will be disappointed. You can take a look at the  magazine  here. In Issue 2 of 1975, the article on pages 3-8 contains a rather complicated statement about the radicals of these numbers.

If $\;\sqrt[n]{a+b\sqrt{d}}=x+y\sqrt{d}$ then $\;\sqrt[n]{a-b\sqrt{d}}=|x-y\sqrt{d}|$

and a sufficient condition for this to happen is

$$\sqrt[n]{a+b\sqrt{d}}=x+y\sqrt{d}\;\;\;\Rightarrow\;\;\sqrt[n]{a^2-b^2\cdot d} \in \mathbb{Q}\tag{N}$$

     The condition (N) is not sufficient, as their example shows

$\sqrt{18+2\sqrt{77}}=\sqrt{11}+\sqrt{7}\neq x+y\sqrt{77}$ although $\sqrt{18^2-2^2 \cdot 77}=4$

                    In my opinion, if $d$ is a prime number, then the condition (N) is indeed

 Necessary_ and_ Sufficient.