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)|

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Section 53 number 4
Computations
The field K =
Q(/2, /3, v5) 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. |G(K/Q)|
4. 1a(Q(/2, /3)|
5. 12(Q(/6)|
Transcribed Image Text:Computations The field K = Q(/2, /3, v5) 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. |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 subgroup
of G(K/F) leaving E fixed. Then À 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 A.
theorem.
53.6 Theorem
1. 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 cosets
of (E) in G(K/F).
5. E is a normal extension of F if and only if )(E) is a normal subgroup of
G(K/F).
hen 1(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.
llu already proved a substantial part of this
Transcribed Image Text: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 subgroup of G(K/F) leaving E fixed. Then À 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 A. theorem. 53.6 Theorem 1. 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 cosets of (E) in G(K/F). 5. E is a normal extension of F if and only if )(E) is a normal subgroup of G(K/F). hen 1(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. llu already proved a substantial part of this
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