Irreducibility over $ \mathbb{Q} ( \sqrt{2} , \sqrt{3})$

Question

Show that $x^5-9 x^3 +15x +6$ is irreducible over $ \mathbb{Q} ( \sqrt{2}, \sqrt{3})$

Answer 1

Some directions:

• Show that $x^5 - 9x^3 + 15x + 6$ is irreducible over $ℚ$. Let $α$ be root of it.
• What can you say about $ℚ(α)$ and $ℚ(√2, √3,α)$?
• What can you say about the minimal polynomial of $α$ over $ℚ(√2,√3)$?