Prove the splitting field of is a radical extension...
when it comes to ,I am a little bit confused. Since many theorem is based on ..
The theorem about determining if something is a radical extension (that is, the polynomial is solvable) is about charachteristic zero fields. However, the definition of radical extensions is more general then that.
Remember, is a radical extension (by definition) iff there exists and such that , .
So construct the splitting field of this and argue that you can write it in the above form.
Notice that is irreducible over . Therefore, as you know, there exists an extension field which has that solves this polynomial. Therefore, . Now . You need to ask now whether has a zero in . Sadly, it does not, this can be confirmed by checking where . Thus, is not the splitting field over . However, we know there exists such that there is which solves . The extension field will therefore become the splitting field over . Now it remains to argue this satisfies the conditions of being a radical extension.