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Subfields of the splitting field of x to the fourth minus seven

Put

α = 47

and

L = Q (α, i)

. The dihedral group

D_{8} = ⟨ r, s : r^{4} = s^{2} = 1, srs = r^{- 1} ⟩

acts by

r (α) = i α, q u a d r (i) = i, qq u a d s (α) = α, q u a d s (i) = - i .

The ten subgroups and their fixed fields are

D_{8} ⟨ r ⟩ ⟨ r^{2}, s ⟩ ⟨ r^{2}, rs ⟩ ⟨ r^{2} ⟩ ⟨ s ⟩ ⟨ r^{2} s ⟩ ⟨ rs ⟩ ⟨ r^{3} s ⟩ {1} Q Q (i) Q (7) Q (- 7) Q (7, i) Q (α) Q (i α) Q ((1 + i) α) Q ((1 - i) α) L .

The normal subgroups are

D_{8}

,

⟨ r ⟩

, the two displayed Klein four-groups,

⟨ r^{2} ⟩

, and

{1}

.

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Solution