# Is this how to find a polynomial with a given splitting field?

Suppose we want to find a polynomial whose splitting field is $$\Bbb Q(\sqrt{2}, \sqrt{-3})$$ over $$\Bbb Q$$. Then the following is how you'd do it right ?

We want to adjoin the roots $$\alpha=\sqrt{2}, \beta=\sqrt{-3}$$. so we could just say $$(x^2-2)(x^2+3)=0$$ and expanding the brackets gives the polynomial .

• Yes, that's exactly how I would do it. – Lord Shark the Unknown Mar 8 at 18:57