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On 2-class field towers for quadratic number fields with 2-class group of type (2,2)
Published online by Cambridge University Press: 18 May 2009
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Let K be a quadratic number field with 2-class group of type (2,2). Thus if Sk is the Sylow 2-subgroup of the ideal class group of K, then Sk = ℤ/2ℤ × ℤ/2ℤ Let
K ⊂ K1 ⊂ K2 ⊂ K3 ⊂…
the 2-class field tower of K. Thus K1 is the maximal abelian unramified extension of K of degree a power of 2; K2 is the maximal abelian unramified extension of K of degree a power of 2; etc. By class field theory the Galois group Ga1 (K1/K) ≅ Sk ≅ ℤ/2ℤ × ℤ/2ℤ, and in this case it is known that Ga(K2/Kl) is a cyclic group (cf. [3] and [10]). Then by class field theory the class number of K2 is odd, and hence K2 = K3 = K4 = …. We say that the 2-class field tower of K terminates at K1 if the class number of K1 is odd (and hence K1 = K2 = K3 = … ); otherwise we say that the 2-class field tower of K terminates at K2. Our goal in this paper is to determine how likely it is for the 2-class field tower of K to terminate at K1 and how likely it is for the 2-class field tower of K to terminate at K2. We shall consider separately the imaginary quadratic fields and the real quadratic fields.
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- Copyright © Glasgow Mathematical Journal Trust 1998
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