ABSTRACT
Let F=ℚ(D)https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq1.tif"/> where D is the discriminant of the field F. Denote by C, the Sylow 2-subgroup of the strict (i.e., narrow) ideal class group of F. If G=g(F/ℚ)https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq2.tif"/> is the Galois group of F/ℚhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq3.tif"/>, and σ is a generating automorphism of G, then there is a natural action of G on C. Further, if x ∈ C and A is an ideal of F representing x, then σx is represented by σA. Since A⋅αA=(NF/ℚ(A))https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq4.tif"/> is a principal ideal generated by a rational number, we see that σx=x−1https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq5.tif"/> for all x ∈ C. In particular, σx=xhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq6.tif"/> if and only if x2=1https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq7.tif"/>, so it follows that CG=C2https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq8.tif"/>, where CG={x|σx=x}https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq9.tif"/> and C2={x|x2=1}https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq10.tif"/>.
