Hilbert's Theorem 90 gives a condition for an algebraic number \beta in a number field k to be a quotient $alpha/\alpha' of conjugate algebraic numbers, for \alpha in a cyclic extension L of k.
In joint work with A. Dubickas, we find a simple necessary and sufficient condition for \beta\in k to be equal to \alpha/\alpha', where now $\alpha$ is unrestricted.
Just as Hilbert's Theorem 90 has an additive version, so our result too has an additive analogue.