I'm working with this problem: Let L/K be a Galois extension with Galois group $S_4.$ Then L is the splitting field of a monic degree 4 irreducible polynomial over K. Char(K)=0.
My method is since [L:K] is finite, there is a primitive element $a. $ Then $a $ has an irrducible polynomial g(x) in K[x]. Since L/K is Galois and irreducible polynomial g(x) has a root $a $ in L, then it splits in L[x]. Then degree of $g(x) = [K(a):K]=[L:K]=|S_4|=24$, not 4. I'm confused. What's wrong? Thank you!