Effective criteria for \(p\)-th powers in a Galois extension
This paper addresses the problem of testing when an algebraic integer \(e\) in a number field \(K\) is a \(p\)-th power (with \(p\) a prime number). The second author gave a criterion for that in his thesis. The paper begins by recalling that criterion. After that it is applied to the \(\chi\)-objects (with \(\chi\) a Galois character corresponding to \(K\)). The main part is section 4, which introduces a trace criterion: given a numerical approximation \(\epsilon_\sigma\) of the conjugate \(\sigma(e)\), the paper gives a device allowing to affirm that \(e\) is in \({K^*}^p\): a weighted sum \(\sum\zeta_\sigma \epsilon_\sigma\) should be near enough from a rational integer: this can be turned easily into a PARI program.
scientific article; zbMATH DE number 1011058