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Funny that Witt is not mentioned here. Indeed, his papers Theorie der quadratischen Formen in beliebigen Körpern (introducing Witt cancellation, Witt decomposition, the Hasse-Witt invariant, the Witt ring and in a little sidestep proving that every quadratic form in $\ge 5$ variables over a $\mathfrak{p}$-adic field is isotropic) and Zyklische Körper und ...

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