84. 1) If B and C are subfields of a field E, then their compositum BVC...

 

Transcribed Image Text:

84. 1) If B and C are subfields of a field E, then their compositum BVC is the intersection of all the subfields of E containing B and C. Prove that if a1,..., an E E, then F(a1) V……v F(an) = F(a1,...,an). (ii) Prove that any splitting field K/F containing B (as in Exercise 83) has the form K = B1v...VB,, where each Bị is isomorphic to B via an isomorphism which fixes F. (Hint: If Gal(K/F) = {ơ1, … ,o,), then define B; = ¡(B).) 85. Using Exercise 84, prove that any splitting field K/F containing a radical extension R/F (as in Exercise 83) is itself a radical extension. Conclude that, in the definition of solvable by radicals, one can assume that the last field B, is a splitting field of some polynomial over F. 84. 1) If B and C are subfields of a field E, then their compositum BVC is the intersection of all the subfields of E containing B and C. Prove that if a1,..., an E E, then F(a1) V……v F(an) = F(a1,...,an). (ii) Prove that any splitting field K/F containing B (as in Exercise 83) has the form K = B1v...VB,, where each Bị is isomorphic to B via an isomorphism which fixes F. (Hint: If Gal(K/F) = {ơ1, … ,o,), then define B; = ¡(B).) 85. Using Exercise 84, prove that any splitting field K/F containing a radical extension R/F (as in Exercise 83) is itself a radical extension. Conclude that, in the definition of solvable by radicals, one can assume that the last field B, is a splitting field of some polynomial over F.