Let $u$ be a real transcendental number over $\mathbb{Q}$ and $\alpha$ a root of $X^2 + u^2 + 1$ in $\mathbb{C}$. We note $K=\mathbb{Q}(u)$ and $E=K(\alpha)$.
I have to show that
elements of $E\setminus\mathbb{Q}$ are transcendental over $\mathbb{Q}$
I considered $x\in E\setminus\mathbb{Q}$ and supposed that $x$ was algebraic over $\mathbb{Q}$.
We get that $\mathbb{Q}(x)$ is a finite extension of $\mathbb{Q}$ but now I feel really stuck.
I tried to find a contradiction by considering $\mathbb{Q}(x)(u)$, $\mathbb{Q}(x)(\alpha)$ and even the fact that $\mathbb{Q}(x)$ is countable but couldn't find anything.
I also tried to find something like $[\mathbb{Q}(x):\mathbb{Q}]=[\mathbb{Q}(x):\mathbb{Q}(u)][\mathbb{Q}(u):\mathbb{Q}]$ so I could have a contradiction since $\mathbb{Q}(u)$ is an $\infty$-dimensional $\mathbb{Q}$ vector space but didn't really succeed.
Can I have an hint :)?