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Splitting field, prime basis
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[QUOTE="PsychonautQQ, post: 5003582, member: 482086"] [B]1. The problem statement, all variafbles and given/known data[/B] 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) [h2]Homework Equations[/h2] 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? [h2]The Attempt at a Solution[/h2] 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? [/QUOTE]
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Splitting field, prime basis
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