Rational Hamilton quaternion algebra

The rational Hamilton quaternion algebra is the associative unital quaternion algebra

B=(-1,-1)_{\mathbb Q}=\mathbb Q\oplus\mathbb Q i\oplus\mathbb Q j\oplus\mathbb Q k,\qquad i^2=j^2=-1,\quad k=ij=-ji.

Its scalars and four coordinates are rational; multiplication is extended bilinearly from these relations.

Division and scalar extension

Conjugation negates the $i,j,k$ coordinates. For $x=a+bi+cj+dk$ , the reduced norm is $x\bar x=a^2+b^2+c^2+d^2$ . It is positive for every nonzero rational coordinate vector, so $x^{-1}=\bar x/(x\bar x)$ ; thus this is a division algebra. Extending scalars to $\mathbb R$ gives Hamilton’s real quaternions.

Integral structures

The Lipschitz order and Hurwitz order are distinct integral subrings whose rational spans equal $B$ . They are not alternative names for this rational algebra.

References

John Voight, Quaternion Algebras, Chapter 11: The Hurwitz order, §11.1, equation (11.1.1), Lemma 11.1.2 and equation (11.1.3); §11.2.