The field K = ℚ(√2,√3,√5)K=Q(2,3,5) is a finite normal extension of ℚ. It can be shown that [K : Q] = 8. Compute |λ(ℚ(√2,√3))| . The notation is that of Theorem 53.6. दी शrem.53.6 Theoremfields E onto the set of all subgroups of G(K/F). The following properties hold for 2:of G(K/F) leaving E fixed. Then A is a one-to-one map of the set of all such intermediate(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 ^(E) be the subgroupaaafields E onto the set of all subgroups of G(KIE), The following properties nold 1or m1. X(E)= G(K|E).2. E = KG(K/E) = Kx(E)•to blat3. For H < G(K|F), ^(KH) = H.** IAE]= |X(E)| and [E : F]= (G(K/F) : ^(E)), the number of left cosetsof 1(E) in G(K /F).%3DE is a normal extension of F if and only if 2(E) is a normal subgroup ofG(K/F). When A(E) is a normal subgroup of G(K|F), then5.amG(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.

Consider the given information: K=ℚ(2,3,5) be the finite normal extension of ℚ. [K:Q]=8 To compute…

Answered: The field K = ℚ(√2,√3,√5)K=Q(2,3,5) is…

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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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

दी श
rem.
53.6 Theorem
fields E onto the set of all subgroups of G(K/F). The following properties hold for 2:
of G(K/F) leaving E fixed. Then A is a one-to-one map of the set of all such intermediate
(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 ^(E) be the subgroup
a
a
a
fields E onto the set of all subgroups of G(KIE), The following properties nold 1or m
1. X(E)= G(K|E).
2. E = KG(K/E) = Kx(E)•
to blat
3. For H < G(K|F), ^(KH) = H.
** IAE]= |X(E)| and [E : F]= (G(K/F) : ^(E)), the number of left cosets
of 1(E) in G(K /F).
%3D
E is a normal extension of F if and only if 2(E) is a normal subgroup of
G(K/F). When A(E) is a normal subgroup of G(K|F), then
5.
am
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.

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