Quaternion

Lore & Background

Hamilton knew that complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. For many years he had known how to add and subtract triples of numbers, but he was stuck on the problem of multiplication and division. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions took shape. He carved the defining formula into the stone of Brougham Bridge: i² = j² = k² = i j k = -1. Hamilton called a quadruple with these rules a quaternion and devoted most of his life to studying and teaching them. After his death, Scottish mathematical physicist Peter Tait became the chief exponent. Quaternions were a mandatory examination topic in Dublin, and there was a professional research association, the Quaternion Association.

Reader's Guide

Quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. Because of their non-commutative multiplication, they do not form a field. They are a special case of a Clifford algebra. According to the Frobenius theorem, the quaternion algebra is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, the last normed division algebra over the real numbers. The unit quaternions give a group structure on the 3-sphere S3 isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). From the mid-1880s, quaternions began to be displaced by vector analysis, developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz, which was conceptually simpler and notationally cleaner. However, quaternions have had a revival since the late 20th century due to their utility in describing spatial rotations. Representations of rotations by quaternions are more compact and quicker to compute than matrices, and unlike Euler angles, they are not susceptible to gimbal lock. They are used in computer graphics, computer vision, robotics, nuclear magnetic resonance image sampling, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics.

Did You Know?

• Quaternion multiplication is not commutative, meaning the order of multiplication affects the result.
• The set of all quaternions is conventionally denoted by H in honor of Hamilton.
• Quaternions are one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers, the other being the complex numbers.

From Hamilton's Numbers to Arbitrary Fields

A quaternion algebra is, at its core, a four-dimensional central simple algebra built over some base field F. The concept lifts Hamilton's original construction—where the coefficient field is the real numbers—into a far more general setting that works over any field whatsoever. In this broader framework, Hamilton's familiar quaternions occupy a special place: they are the unique quaternion algebra over the reals that is not simply the algebra of 2×2 real matrices. Over the complex numbers, the analogous object is the algebra of biquaternions. One of the defining features of any quaternion algebra is that, once you extend scalars to a suitable larger field K, the resulting tensor product becomes isomorphic to the 2×2 matrix algebra over K. This property—becoming a matrix algebra after a field extension—is what makes quaternion algebras a natural and elegant generalization of the concrete number system Hamilton devised in the nineteenth century, while simultaneously opening the door to deep questions in algebraic number theory and the theory of central simple algebras.

Basis, Multiplication, and the Division-Split Dichotomy

When the base field F has characteristic different from two, every quaternion algebra admits a concrete description as a four-dimensional vector space spanned by the symbols 1, i, j, and k. Two parameters a and b, chosen from the nonzero elements of F, govern the multiplication: i squared equals a, j squared equals b, the product ij yields k while ji yields negative k, and from these rules one derives that k squared equals negative ab. The two most familiar instances over the reals arise from specific choices: setting both a and b to negative one recovers Hamilton's quaternions, whereas taking a equal to negative one and b equal to positive one produces the split-quaternions, in which k squared becomes positive one and the cross-product jk equals negative i. Algebraists denote such a construction as (a,b) over F. A quaternion algebra is either a division algebra—where every nonzero element has a multiplicative inverse—or it is isomorphic to the 2×2 matrix algebra, in which case it is called split. The boundary between these two possibilities is detected by the norm form: the algebra is a division algebra precisely when the associated quadratic form is anisotropic, meaning it vanishes only at the zero element. Equivalently, the conic defined by ax² + by² = z² has no nontrivial solution in F.

Quadratic Forms and the Brauer Group

Quaternion algebras sit at a remarkable crossroads of number theory and algebraic structure. They serve as concrete, computable objects that generate the elements of order two in the Brauer group of a field F, a group that classifies central simple algebras up to Morita equivalence. For important families of fields, including the algebraic number fields, every element of order two in the Brauer group can be represented by a single quaternion algebra. More generally, a deep theorem of Alexander Merkurjev guarantees that for any field, every order-two Brauer group element decomposes as a tensor product of quaternion algebras. In the local setting of p-adic fields, the construction of quaternion algebras takes on an especially transparent form: it can be identified with the quadratic Hilbert symbol that appears in local class field theory. This connection makes quaternion algebras indispensable tools for studying quadratic forms, understanding the arithmetic of fields, and bridging the gap between explicit algebraic constructions and the abstract invariants of class field theory.

Frobenius's Theorem and the p-Adic Surprise

A classical result of Frobenius establishes that over the real numbers there are exactly two quaternion algebras up to isomorphism: the algebra of 2×2 real matrices and Hamilton's quaternions themselves. A parallel statement holds over any local field: precisely two quaternion algebras exist, one split and one a division algebra. Yet the division algebra over a local field is not automatically Hamilton's construction. Over the p-adic numbers, for instance, Hamilton's quaternions form a division algebra only when p equals two. For every odd prime p, the p-adic version of Hamilton's quaternions collapses to the 2×2 matrix algebra. The reason is elegant: the congruence x² + y² = −1 modulo p is solvable for odd p, and Hensel's lemma lifts this solution to the p-adic numbers. Consequently, the quaternion xi + yj + k has norm zero and therefore lacks a multiplicative inverse, proving the algebra is split. This single observation, rooted in a simple modular arithmetic fact, illustrates how local arithmetic can dramatically alter the algebraic structure of what appears, over the reals, to be a fixed and immutable object.