1. The problem statement, all variafbles and given/known data Problem: Let E be a splitting field of f over F. If [E:F] is prime, show that E=F(u) for some u in E (show that E is a simple extension of F) 2. Relevant equations Things that might be useful: If E>K>F are fields, where K and F are subfields of E and F is a subfield of K, then [E:F] = [E:K][K:F] since E is a splitting field of f: f = a(x-(u1))(x-(u2)).......(x-(up)) E = F(u1,u2,.....,up) Did i write this correctly in the sense that if [E:F] is prime and E is a splitting field of f then f will have p roots in E? 3. The attempt at a solution My most promising method of proving this is using the multiplication theorem stated above, noting that E = F(u1,u2....up) so... p = [E:F] = [F(u1,u2....up):F(u)] * [F(u1),F) since p is prime, this would force [F(u1,u2....up):F(u1)] to equal one and so F(u1,u2....up) = F(u1). I realize this isn't the most thorough argument, and possibly just straight up incorrect. Anybody that knows what they're talking about have any comments? Am I on the right track?