The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.

|λ(Q(√2, √3))|

Data from 53.6 Theorem

(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 λ(E) be the subgroup of G(K/F) leaving E fixed. Then λ is a one-to-one map of the set of all such intermediate fields E onto the set of all subgroups of G(K/F). The following properties hold for λ: 

1. λ(E) = G(K/E).  

2. E = K_G(K/E) = K_λ(E). 

3. For H ≤ G(K/F),λ(E_H) = H. 

4.  [K : E] = |λ(E)] and [E : F] = (G(K/F) : λ(E)), the number of left cosets of λ(E) in G(K/F). 

5. E is a normal extension of F if and only if λ(E) is a normal subgroup of G(K/F). When λ(E) is a normal subgroup of G(K/F), then G(E/F) ≈ G(K/F)/G(K/E). 

6. The diagram of subgroups of G(K/F) is the inverted diagram of intermediate fields of K over F.