Introduction to Quadratic Forms over Fields

anisotropic anisotropic form anisotropic over F assume binary form Chapter clearly closed field construction Corollary CSGA defined denote dim q division algebra element equation equivalent euclidean exists extension K/F extension of F F is formally F is nonreal F-algebra F-form F/F² fact field extension field F finite field form over F form q formally real field function field Galois graded algebra hence homomorphism hyperbolic implies induction integer isometry isomorphism Lemma Let F multiplicative n-fold Pfister form nonreal fields nonzero notation number field ordering orthogonal Pfister form Pfister neighbor polynomial prime ideal Proof Proposition prove Pythagoras number pythagorean field q is isotropic quadratic extension quadratic form theory quadratic forms quadratic space quadratically closed quaternion algebras real-closed field result splits square classes squares in F subgroup Theorem torsion u-invariant unique vector Witt rings write Wt(F zero