Section 53 number 4 ComputationsThe field K =Q(/2, /3, v5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. In Exercises 1through 8, compute the indicated numerical quantity. The notation is that of Theorem 53.6.2. |G(K/Q)|4. 1a(Q(/2, /3)|5. 12(Q(/6)| then give another example, and finally, com-(Main Theorem of Galois Theory) Let K be a finite normal extension of a field F.with Galois group G(K/F). For a field E. where F < E < K, let X(E) be the subgroupof G(K/F) leaving E fixed. Then À is a one-to-one map of the set of all such intermediatefields E onto the set of all subgroups of G(K|F). The following properties hold for A.theorem.53.6 Theorem1. X(E) = G(K|E).2. E = KG(K|E) = KxE»•3. For H < G(K/F), \(KH) = H.4. [K : E] = |1(E)| and [E : F] = (G(K/F) : (E)), the number of left cosetsof (E) in G(K/F).5. E is a normal extension of F if and only if )(E) is a normal subgroup ofG(K/F).hen 1(E) is a normal subgroup of G(K|F), thenG(E/F)~ G(K|F)/G(K/E).6. The diagram of subgroups of G(K/F) is the inverted diagram of intermediatefields of K over F.llu already proved a substantial part of this

Name: Section 53 number 4 ComputationsThe field K =Q(/2, /3, v5) is a finite
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Description: Section 53 number 4 ComputationsThe field K =Q(/2, /3, v5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. In Exercises 1through 8, compute the indicated numerical quantity. The notation is that of Theorem 53.6.2. |G(K/Q)|4. 1a(Q

The field K=ℚ2,3,5 is a finite normal extension of ℚ To find the numerical quantity λℚ2,3 According…

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Computations The field K =Q(/2, /3, /5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. In Exercises 1 through 8, compute the indicated numerical quantity. The notation is that of Theorem 53.6. 2. IG(K/Q)| 4. 1A(Q(/2, /3)|