Hilbert's Theorem 90
Hilbert's Theorem 90 is the surprising fact that for a nice field extension, every element of norm one is secretly a ratio of an element to its conjugate. In its original form it concerns a cyclic extension with generator σ: if an element a has norm 1 (the product of all its conjugates equals 1), then a must equal b/σ(b) for some b. It looks like a special equation-solving trick, but it is the visible face of a cohomology group vanishing.
The cohomological statement is: for any finite Galois extension L/K with group G, the first cohomology of the multiplicative group vanishes, H^1(G, L^*) = 0. Unwinding the definition, a 1-cocycle is a map g |-> a_g in L^* with a_{gh} = a_g · g(a_h), and vanishing means every such cocycle is principal: a_g = g(b)/b for some b in L^*. The classical norm-one statement is the special case where G is cyclic generated by σ and the cocycle is determined by a_σ = a.
The theorem is the load-bearing wall of Kummer theory and of the description of cyclic extensions: combined with the long exact sequence from 1 -> μ_n -> L^* -> L^* -> 1 (the n-th power map), it gives K^*/(K^*)^n ≅ H^1(G, μ_n), classifying cyclic extensions of degree n when K contains the n-th roots of unity. An additive analogue, the additive Hilbert 90, states H^1(G, L) = 0 and is even simpler, following from the normal basis theorem.
Take L = Q(i), K = Q, G = {1, σ} with σ complex conjugation. An element a + bi has norm a^2 + b^2; if this equals 1 then a + bi = b'/σ(b') for some b' in L^*. For instance 3/5 + 4i/5 (norm 1) equals (2 + i)/σ(2 + i) = (2 + i)/(2 - i).
A norm-one Gaussian rational written as b/σ(b).
The name comes from its being the 90th theorem in Hilbert's 1897 report Zahlbericht; the cohomological generalization is due to Emmy Noether. The multiplicative vanishing is special — H^2(G, L^*) is generally nonzero and gives the Brauer group.