## Abstract

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 2^{n} 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/2^{n}ℤ of order 2^{n} with *n* ∈ ℕ, resp. , a (ℤ/2^{n}ℤ)-extension resp. a -extension of *K*.

A quadratic extension *L*|*K* is called (ℤ/2^{n}ℤ)-embeddable, resp. -embeddable, if there exists a (ℤ/2^{n}ℤ)-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

*1*

^{m−1}−*,*2

*1*

^{m}−*or*2

*1*

^{m+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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