Let $k$ be a field of characteristic $\ne 2$ and $L$ be a finite extension of the function field $k(t)$ in which $\sqrt{t}$ and $\sqrt{1+t}$ exist.
Find the minimal polynomial of $T:=\sqrt{t}+\sqrt[]{1+t}$.
Some calculations show that $P(T)=0$ where $P(T):= T^4-2(2t+1)T^2+1$.
I am trying to use Eisenstein criterion to prove that P is irreducible over $k(t)[X]$.
Since the constant coefficient of $P(T)$ is $1$, I can't apply Eisenstein criterion, so I computed $P(T+1)=T^4+4T^3+4(1-t)T^2-8tT-(4t+1)$ and I still have a problem with the constant term.
Thank you for any help or hints to prove irreducibility of P over $k(t)$ even without Eisenstein's criterion.
Answer based on reuns'comments as it might be helpful to someone and I hope it is error-free:
$1+t$ is irreducible over $k[t]$ as it is of degree 1 and it verifies $(1+t)\ |-(1+t)$ and $(1+t) \nmid 1$. By Eisenstein criterion, $X^2-(1+t)$ is irreducible over $k(t)$.
$t=(\sqrt[]{t+1})^2-1=(\sqrt[]{t+1}-1)(\sqrt[]{t+1}+1)$ and $(\sqrt[]{t+1}+1) $ is irreducible over $k[t,\sqrt[]{t+1}]$ and $(\sqrt[]{t+1}+1) \nmid 1$ so By Eisenstein criterion, $X^2-t$ is irreducible over $k(t,\sqrt[]{t+1})$.
We deduce that $\pm \sqrt[]t \pm \sqrt[]{t+1}$ are conjugates over $k(t)$ and that $X^4-2(2t+1)+1=((X-\sqrt{t})^2-(1+t))((X+\sqrt{t})^2-(1+t))$ is irreducible over $k(t)$.