Calculate the degree $[K:\mathbb{Q}]$

Question

Let $a\in \mathbb{C}$,$a^2=\frac{3+i\sqrt{7}}{2}$ and $K=\mathbb{Q}(a)$.How can we calculate the extension degree $[K:\mathbb{Q}]$ without using minimal polynomials.Its obvious that $a$ is root of $f(x)=x^4-3x^2+4$ but how can we prove it without search if $f(x)$ is irreducible?

Answer 1

This amounts to proving that $ (3 + \sqrt{-7})/2 $ can't be a perfect square in $ \mathbf Q(\sqrt{-7}) $. An easy argument to see this is that the square of $ (1 - \sqrt{-7})/2 $ is equal to $ (-3 - \sqrt{-7})/2 $, so if $ (3 + \sqrt{-7})/2 $ were a perfect square then their ratio $ -1 $ would also be a square in $ \mathbf Q(\sqrt{-7}) $. This would imply $ \mathbf Q(\sqrt{-1}) = \mathbf Q(\sqrt{-7}) $, which is easily seen to be false.