Abstract Algebra; Galois field extension and automorphisms

Question

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 ? :)

Answer 1

Hint:

Solve first for $8+2\sqrt 7$.

Answer 2

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 $$ \sqrt{4+\sqrt{7}} = \sqrt{\frac72}+\sqrt{\frac12} $$