# additive form of Hilbert’s theorem 90

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

Title additive form of Hilbert’s theorem 90 AdditiveFormOfHilbertsTheorem90 2013-03-22 15:20:35 2013-03-22 15:20:35 mathwizard (128) mathwizard (128) 7 mathwizard (128) Theorem msc 11R32 msc 12F10 HilbertTheorem90