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Let $K/k$ be an extension of number fields. We describe theoretical resultsand computational methods for calculating the obstruction to the Hasse normprinciple for $K/k$ and the defect of weak approximation for the norm one torus$R^1_{K/k} \mathbb{G}_m$. We apply our techniques to give explicit andcomputable formulae for the obstruction to the Hasse norm principle and thedefect of weak approximation when the normal closure of $K/k$ has symmetric oralternating Galois group.

Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensions