Let m be an integer such that m≥7 and m≡0,1,7mod8. We construct strictly compatible systems of representations Γ ℚ →Spin (ℚ ¯ l )→ spinGL N ) are potentially automorphic motivic. As application, we prove instances the inverse Galois problem for

Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give method that completely describes the factorization patterns of linear resolvent $f(x)$ using simple arithmetic conditions on $a$ and $b$. As result, we determine exact six possible Galois groups classify all them. application, characterize polynomials $x^8+ax^4+1 \mathbb{Q}[x]$. also use similar ...