<s>[INST] Abstract Algebra; Galois field extension and automorphisms I am really curious...
Can $\sqrt{4+\sqrt{7}}$ be written in the form $\sqrt a + \sqrt b $ with rational numbers $a$ and $b$ ?
I was thinking that we could try to brute force it by equating $\sqrt{4+\sqrt{7}} = x$ and then try to work out the $a$ and $b$ ? Any thoughts ? :)
 [/INST] Wikipedia describes a simple method to write
$$
\sqrt{a+b \sqrt{c}\ } = \sqrt{d}+\sqrt{e}
$$
with
$$
d=\frac{a + \sqrt {a^2-b^2c}}{2},
\qquad
e=\frac{a - \sqrt {a^2-b^2c}}{2}
$$
This works iff $a^2 - b^2c$ is a square.
For $\sqrt{4+\sqrt{7}}$ we have $a^2 - b^2c=9$ and so
$$
\sqrt{4+\sqrt{7}} = \sqrt{\frac72}+\sqrt{\frac12}
$$
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