1. The problem statement, all variables and given/known data I'm following the solutions to a homework tutorial and I'm having trouble understanding why what they're saying is true. Question: Let f be a polynomial in K[x] and let S be the splitting field of f over K. decide whether the extension S:K is galois and describe the structure of the Galois group Gal(S:K) for the polynomial x^3+2x+1 over ##F_3##. 2. Relevant equations 3. The attempt at a solution S:F_3 will be a galois extension because the polynomial is separable and S is a finite splitting field for f over F_3. Now I tried to construct the splitting field of F_3 to figure out what S looks like by first noticing that f has no roots in F_3, I then adjoined a root r to F_3. My notes say that "F_3(r):F_3 is normal and so all the roots of the minimal polynomial are in F_7(r)." I'm quite confused on how they know this extension is normal. I get that S:F is Galois, but for any intermediate field K such that S>K>F the extension K>F is not gaurentee'd to be normal. How do they know F_3(r):F_3 is normal?