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  "body": "\n## Statement\n\n1. Let $L/K$ be a finite [[cyclic extension|VFE5B71E7Y]] and let $\\sigma$ be a generator of its [[Galois group|0XVR72M51M]].\n   An element $\\alpha \\in L$ has [[norm|8HHY1K3XDC]] $N_{L/K}(\\alpha) = 1$ if and only if $\\alpha = \\beta / \\sigma(\\beta)$ for some $\\beta \\in L^\\times$.\n2. Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]] with Galois group $G$.\n   Let $f \\colon G \\to L^\\times$ be a map with $f(\\sigma\\tau) = f(\\sigma)\\, \\sigma(f(\\tau))$ for all $\\sigma, \\tau \\in G$.\n   Then there is $\\beta \\in L^\\times$ with $f(\\sigma) = \\beta / \\sigma(\\beta)$ for all $\\sigma \\in G$.\n   In the language of group cohomology, $H^1(G, L^\\times) = 1$.\n\n## Remarks\n\nFor part 2, [[Dedekind's lemma|N43G50YHG2]] gives $\\gamma \\in L$ such that $\\beta = \\sum_{\\tau \\in G} f(\\tau)\\, \\tau(\\gamma)$ is nonzero.\nThen $\\sigma(\\beta) = \\sum_\\tau \\sigma(f(\\tau))\\, \\sigma\\tau(\\gamma) = f(\\sigma)^{-1} \\sum_\\tau f(\\sigma\\tau)\\, \\sigma\\tau(\\gamma) = f(\\sigma)^{-1} \\beta$.\nFor part 1, if $N(\\alpha) = 1$ then $f(\\sigma^i) = \\alpha\\, \\sigma(\\alpha) \\cdots \\sigma^{i-1}(\\alpha)$ is well defined on the cyclic group and satisfies the condition of part 2; conversely $N(\\beta / \\sigma(\\beta)) = N(\\beta)/N(\\sigma(\\beta)) = 1$.\n\nExample: take $L = \\Q(i)$ and $K = \\Q$, with $\\sigma$ complex conjugation.\nThe elements of norm $1$ are the $a + bi$ with $a^2 + b^2 = 1$, and the theorem writes each as $(m + ni)/(m - ni) = (m^2 - n^2 + 2mni)/(m^2 + n^2)$ with integers $m, n$ not both zero.\nThis recovers the parametrization of rational points on the unit circle and of Pythagorean triples.\n\nPart 1 needs a generator $\\sigma$ of the Galois group, so it applies only to cyclic extensions; for a general finite Galois extension one uses part 2.\n\n## Explanations\n\n### L1 human\n\nThe norm of an element multiplies it with all its conjugates.\nHilbert's Theorem 90 says that, in a cyclic extension, the elements of norm $1$ are exactly the quotients of an element by its conjugate.\nFor the Gaussian numbers this explains why every rational point on the circle $a^2 + b^2 = 1$ has the form $\\left(\\frac{m^2 - n^2}{m^2 + n^2}, \\frac{2mn}{m^2 + n^2}\\right)$.\n\n### L2 human\n\nThe cohomological form $H^1(\\Gal(L/K), L^\\times) = 1$ is the starting point of [[Galois cohomology|9MPCZ3EAJM]].\nIt is the main step in Kummer theory.\nWhen $K$ contains a primitive $n$-th root of unity, it shows that every cyclic extension of degree $n$ is obtained by adjoining an $n$-th root.\nThere is also an additive version: in a cyclic extension, the elements of [[trace|WPE5Z53QYJ]] $0$ are exactly the $\\beta - \\sigma(\\beta)$, which gives Artin–Schreier theory in characteristic $p$.\n",
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