Additive Hilbert's Theorem 90 in the ring of algebraic integers

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For a number field K , we give a complete characterization of algebraic numbers which can be expressed by a difference of two K -conjugate algebraic integers. These turn out to be the algebraic integers whose Galois group contains an element, acting as a cycle on some collection of conjugates which sum to zero. Hence there are no algebraic integers which can be written as a difference of two conjugate algebraic numbers but cannot be written as a difference of two conjugate algebraic integers. A generalization of the construction to a commutative ring is also given. Furthermore, we show that for n ⩾_ 3 there exist algebraic integers which can be written as a linear form in n K -conjugate algebraic numbers but cannot be written by the same linear form in K -conjugate algebraic integers.