Splitting field, prime basis

  • #1
784
11
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?
 

Answers and Replies

  • #2
18,363
8,215
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?
 

Related Threads on Splitting field, prime basis

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
8K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
13
Views
4K
  • Last Post
Replies
16
Views
4K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
5K
Replies
0
Views
1K
  • Last Post
Replies
5
Views
626
Top