TITLE: Applications of additive version of Hilbert's theorem 90
QUESTION [6 upvotes]: Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$,  and $a$ is element of $L$ with trace 0, there exists an element $b$ of $L$ such that $a = b - g(b)$.
The corresponding multiplicative version with norm instead of trace has many interesting applications, for instance determining Pythagorean triples, or solving Pell's equation. How about the additive version? Where it can be useful?

REPLY [6 votes]: Google Artin-Schreier theory, an additive analogue of Kummer theory. In particular, the proof of the Artin-Schreier theorem about non-algebraically closed fields whose algebraic closure is a finite extension is a beautiful application (when treating the case of fields with characteristic $p$).
Quite generally, these additive theorems are useful in characteristic $p$ when dealing with extensions of degree $p$ or more generally a power of $p$, depending on the situation. If you are not a fan of characteristic $p$ then the additive Hilbert Theorem 90 will not be your friend.