Question: Why is this Extension Normal?

[PsychonautQQ]

Homework Statement

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 $F_3$.

Homework Equations

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?

 

[fresh_42]

Homework Statement

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 $F_3$.

Homework Equations

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?

Why $\mathbb{F}_7$? You change the characteristic this way. If you have a root $r$, why don't you calculate the other two roots of $f(x)=x^3+2x+1$?

 

[PsychonautQQ]

I'm sorry, I don't know why I keep typing $F_7$ >.<