On the arithmetic of quaternion algebras

Abstract

In this section, F is a field of characteristic = 2. Unless stated otherwise, all algebras considered here are finite dimensional algebras over F . If 1 A (or simply 1) is the identity of an F -algebra A, then the map α → α1 A is a monomorphism of F -algebras. This map identifies F as a subalgebra of A. If R is a ring, then R × denotes the group of units in R. 1.1 Basic Definitions Definition 1.1 A quaternion algebra H over F is a 4-dimensional algebra over F with a basis {1, i, j, k} such that i 2 = a, j 2 = b, ij = k = −ji for some a, b ∈ F × . In this definition, notice that k 2 = −ab. The basis {1, i, j, k} is called a standard basis for H and we write H = a,b F . Note that there are infinitely many standard bases for H, and hence there could be another pair of nonzero elements c, b ∈ F , different from the pair a, b, such that c,d F = a,b F . For instance, a,b F = ax 2 ,by 2 F for any x, y ∈ F × , and a,b F = a,−ab F . The notation H = a,b F is functorial in F , that is, if K is a field extension of F , then a, b F ⊗ K ∼ = a, b K as K-algebras. Example 1.2 In M 2 (F), let i = 1 0 0 −1 , j = 0 1 1 0 , k = ij = 0 1 −1 0 . Then i 2 = j 2 = 1 and {1, i, j, k} is a basis for M 2 (F). Therefore, M 2 (F) = 1,1 F = 1,−1 F . Example 1.3 Another familiar example of quaternion algebras is Hamilton's quaternions H. It is a quaternion algebra over R with a basis {1, i, j, k} such that i 2 = −1, j 2 = −1, ij = k = −ji. This shows that H = −1,−1 R . A simple calculation shows that any two elements from {i, j, k} are anti-commutative. Moreover, ij = k, jk = i and ki = j.