$\gcd$ of polynomials over $F_p$

Question

Let $p$ be an odd prime number and $K:=F_{p^{t}}$ be a field of characteristic $p$. Let $u\in K-F_p$. Prove that for all $n\ge1$ $\gcd(T^{2^{n}−2}+T^{2^{n}−4}+T^{2^{n}−8}+⋯+T^{2^{n}−2^{n−1}}+1,T−u)=1?$