# Splitting field, prime basis

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)

## Homework 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?

## 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?