Finding a minimal polynomial over $\mathbb{Q}(\sqrt5)$

Question

The question is to find the minimal polynomial of $\sqrt{2}+\sqrt{7}$ over $\mathbb{Q}(\sqrt{5})$.

First, I found its minimal polynomial over $\mathbb{Q}$ which is equal to $X^4 - 18X^2 + 25$. I suppose this could already be a candidate for a minimal polynomial over $\mathbb{Q}(\sqrt{5})$ so I tried proving that using the tower property, but I don't think that's the right approach.

Answer 1

Hint:

First note that the minimal polynomial must divide $x^4-18x^2+25$ and using the tower law we can say that if this is not the minimal polynomial then it has to be quadratic.

If you know what the other roots of $x^4-18x^2+25$ are then you can check all 6 pairs (in fact you only need 3 of them), to see if they have coefficients in $\mathbb{Q}(\sqrt{5})$ or not.

There is a quicker method using the Galois group, but I'm guessing you haven't met this yet?