Abstract:Let k be a field, T be an algebraic k-torus, X is a smooth k-compactification of T and Pic X is the Picard group of X = X × k k. Hoshi, Kanai and Yamasaki [HKY] determined H 1 (k, Pic X) for norm one tori T = R(1) K/k ( m) and gave a necessary and sufficient condition for the Hasse norm principle for extensions K/k of number fields with [K : k] = n ≤ 15 and n = 12. In this paper, we determine 64 cases with H 1 (k, Pic X) = 0 and give a necessary and sufficient condition for the Hasse norm principle for K/k whe… Show more
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