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Embeddability of quadratic extensions in cyclic extensions
For an algebraic number field K we study the quadratic extensions of K which can be embedded in a cyclic extension of K of degree 2n for all natural numbers n, as well as the quadratic extensions which can be embedded in an infinite normal extension with the additive group of 2-adic integers as Galois group.
For shortness we call a normal extension of K whose Galois group is the cyclic group ℤ/2nℤ of order 2n with n ∈ ℕ, resp. , a (ℤ/2nℤ)-extension resp. a -extension of K.
A quadratic extension L|K is called (ℤ/2nℤ)-embeddable, resp. -embeddable, if there exists a (ℤ/2nℤ)-extension, resp. a -extension, of K containing L.
One main result of this paper is the following observation, the exact formulation of which is given in theorems 6 to 8 in §3:
Theorem 0. Let K be an imaginary quadratic number field whose discriminant has m prime divisors. Then the number of quadratic extensions L|K which are (ℤ/2 n ℤ)-embeddable for all n is 2 m−1 − 1, 2 m − 1 or 2 m+1 − 1, depending on certain congruences for the discriminant and its prime divisors. But the number of quadratic extensions L|K which are -embeddable is only 3.
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