08-12-2026, 01:01 PM
Quaternion algebra
Summary
A quaternion algebra is a four-dimensional algebra over a field $F$ that generalizes Hamilton’s quaternions and is an important object in abstract algebra and number theory. When the characteristic of $F$ is not $2$, it can be described using a basis ${1,i,j,k}$ with multiplication rules $i^2=a$, $j^2=b$, $ij=k$, and $ji=-k$, where $a,b\in F$. Every element has the form $q=x_0+x_1i+x_2j+x_3k$, and the algebra is denoted $\left(\frac{a,b}{F}\right)$.
Its norm is the quadratic form $N(q)=x_0^2-a x_1^2-b x_2^2+ab x_3^2$, which plays an important role in determining whether the algebra is a division algebra or is split. Quaternion algebras are closely connected with quadratic forms, the Brauer group, Hilbert symbols, and number theory; over the rational numbers, they can be classified by the places at which they ramify.
ARTICLE / ARCHIVE
Summary
A quaternion algebra is a four-dimensional algebra over a field $F$ that generalizes Hamilton’s quaternions and is an important object in abstract algebra and number theory. When the characteristic of $F$ is not $2$, it can be described using a basis ${1,i,j,k}$ with multiplication rules $i^2=a$, $j^2=b$, $ij=k$, and $ji=-k$, where $a,b\in F$. Every element has the form $q=x_0+x_1i+x_2j+x_3k$, and the algebra is denoted $\left(\frac{a,b}{F}\right)$.
Its norm is the quadratic form $N(q)=x_0^2-a x_1^2-b x_2^2+ab x_3^2$, which plays an important role in determining whether the algebra is a division algebra or is split. Quaternion algebras are closely connected with quadratic forms, the Brauer group, Hilbert symbols, and number theory; over the rational numbers, they can be classified by the places at which they ramify.
ARTICLE / ARCHIVE
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