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Exercise 5.2.5 Consider the subfield Q(√2, √√3) CR. a. Prove that [Q(√2, √3): Q] =4 by proving that [Q(√2): Q] = 2, and
Posted: Thu Jul 07, 2022 2:20 pm
by answerhappygod
- Exercise 5 2 5 Consider The Subfield Q 2 3 Cr A Prove That Q 2 3 Q 4 By Proving That Q 2 Q 2 And 1 (17.14 KiB) Viewed 63 times
Exercise 5.2.5 Consider the subfield Q(√2, √√3) CR. a. Prove that [Q(√2, √3): Q] =4 by proving that [Q(√2): Q] = 2, and that ²-3 is irreducible in Q(√2)[z], then appeal to Proposition 5.2.6 and Exercise 5.2.4. Hint: to prove that 22 -3 is irreducible, suppose there is a+b√2 € Q(√2) so that (a+b√2)2-3-0 and derive a contradiction. b. Prove that 1, √2, √3, √6 is a basis for Q(√2, √3) over Q. c. Prove that Aut(Q(√2, √3), Q) = Z₂ x Z₂2. Hint: Aut(Q(√2, √3), Q) acts on the roots (√2,-√2, √3,-√3) of -2 and ²-3.