On Tate-Shafarevich groups of .
Why is the class number of even?
In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.
Arithmetic of Pell surfaces
The class number one problem for some non-abelian normal CM-fields of degree
We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to , the alternating group of degree and order . There are two such fields with Galois group (see Theorem 14) and at most one with Galois group SL (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number .
On the family of Thue equations x³ - (n-1)x²y - (n+2)xy² - y³ = k
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