# Splitting field, prime basis

1. Feb 9, 2015

### PsychonautQQ

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?

2. Feb 14, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?