Topology of Numbers · Problem 8.4.3

Quadratic Fields — Allen Hatcher

Mathematics — Number Theory · ideal-class-group

Problem

In Z[−14]\mathbb{Z}[\sqrt{-14}] do the following: (a) Compute the square of the ideal (2,−14)(2, \sqrt{-14}). (b) For the ideal L=(3,1+−14)L = (3, 1 + \sqrt{-14}) find a reduced basis for L2L^2 and use this to draw a picture of the lattice L2L^2. (c) Find nonzero elements α\alpha and β\beta in Z[−14]\mathbb{Z}[\sqrt{-14}] such that αL2=β(2,−14)\alpha L^2 = \beta(2, \sqrt{-14}).

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