On the shapes of pure prime-degree number fields

For p prime and ℓ=p-1 2, we show that the shapes of pure prime degree number fields lie on one of two ℓ-dimensional subspaces of the space of shapes, and which of the two subspaces is dictated by whether or not p ramifies wildly. When the fields are ordered by absolute discriminant we show that the shapes are equidistributed, in a regularized sense, on these subspaces. We also show that the shape is a complete invariant within the family of pure prime degree fields. This extends the results of Harron, in [15], who studied shapes in the case of pure cubic number fields. Furthermore we translate the statements of pure prime degree number fields to statements about Frobenius number fields with a fixed resolvent field. Specifically we show that this study is equivalent to the study of F p -number fields, F p =C p ⋊C p-1 , with fixed resolvent field ℚ(ζ p ).