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Quadratic Forms of Type E6, E7 and E8
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This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.
Princeton University Press
Title: Quadratic Forms of Type E6, E7 and E8
Description:
This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈.
It first recalls the definition of a quadratic space Λ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8.
If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K.
Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈.
The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.
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