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Octonion algebras are certain algebras with a multiplicative quadratic form.In their 2019 article, Alsaody and Gille show that, for octonion algebras overunital commutative rings, there is an equivalence between isotopes andisometric quadratic forms. The contravariant equivalence from unitalcommutative rings to affine schemes, sending a ring to its spectrum, leads usto a question: can the equivalence of isotopy and isometry be generalized tooctonion algebras over a (not necessarily affine) scheme? We present the basicdefinitions and properties of octonion algebras, both over rings and overschemes. Then we show that an isotope of an octonion algebra C over a scheme isisomorphic to a twist by an Aut(C)-torsor. We conclude the thesis by giving anaffirmative answer to our question.

Octonion Algebras over Schemes and the Equivalence of Isotopes and Isometric Quadratic Forms