Definite quaternion algebra

Let $K$ be a totally real number field, meaning that every embedding $K\hookrightarrow\mathbb C$ lands in $\mathbb R$ . A quaternion algebra $B/K$ is totally definite if it is ramified at every real place. It is indefinite if it splits at at least one real place.

Over $\mathbb Q$ , there is one real place: “definite” means $B\otimes_{\mathbb Q}\mathbb R\cong\mathbb H$ , and “indefinite” means $B\otimes_{\mathbb Q}\mathbb R\cong M_2(\mathbb R)$ .

Norm interpretation

In the definite rational case, the reduced norm on the real scalar extension is a positive-definite quadratic form. The rational Hamilton algebra is the basic example. In the split real algebra, determinant is not positive definite.

Scope of the convention

The definition above fixes a totally real base. Having no real places, as for an imaginary quadratic field, does not make an algebra totally definite by a vacuous condition: complex places are split. Equivalently, a totally definite quaternion algebra over an arbitrary number field must ramify at all archimedean places, which forces the field to be totally real.

References

John Voight, Quaternion Algebras, author's text, Definition 14.5.7 and §14.5.8.