Definition:Represented by Quadratic Form

Definition

Let $R$ be an integral domain.

Let $M$ be an $R$-module.

Let $q : M \to R$ be a quadratic form.

Let $a\in R$.

Then $q$ represents $a$ (over $R$) if and only if there exists $x \in M \setminus \set 0$ with $\map q x = a$.

Also defined as

Some authors do not require the given preimage of $a$ to be nonzero. The only difference is, with that convention, that every quadratic form represents $0$, whereas such forms are otherwise known as isotropic forms.

Also see

• Definition:Equivalent Quadratic Forms
• Definition:Isotropic Quadratic Form

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