additive form of Hilbert’s theorem 90

Let $L/K$ be a finite cyclic Galois extension and $\sigma$ a generator of $\mathrm{Gal}(L/K)$. An element $y\in L$ satisfies $\mathrm{Tr}(y)=0$ if and only if there exists $x\in L$ satisfying $y=x-\sigma (x)$. Here $\mathrm{Tr}$ denotes the trace in $L/K$.