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 […]

First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has $\sigma(\tau(a))=(\sigma\tau)(a), \sigma(ab)=\sigma(a)\sigma(b)$. A 1-cocycle from $G$ to $A$ is a function $f: G\to A$ such that $f(\sigma\tau)=f(\sigma)\sigma(f(\tau))$ for any $\sigma, \tau\in G$. Two […]

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