Example

— 1) Take $K = \mathbf{R}$, $A = \mathbf{C}$. Taking into account Alg., chap. VIII, §12, n° 2, prop. 4, we obtain $\mathrm{mod}_\mathbf{C}(z) = |z|^2$ for all $z \in \mathbf{C}$. (2)

2) Take $K = \mathbf{R}$, and for A the quaternion field $\mathbf{H}$ (GT, VIII, §1, No. 4). Consider the following elements of $\mathbf{M}_2(\mathbf{C})$:

which, together with $I_2$, form a basis of $\mathbf{M}_2(\mathbf{C})$ over $\mathbf{C}$. One verifies easily that

The mapping $a + b i + c j + d k \mapsto a I_2 + b X_1 + c X_2 + d X_3$ may therefore be extended to a $\mathbf{C}$-isomorphism of the algebra $\mathbf{C} \otimes_\mathbf{R} \mathbf{H}$ onto the algebra $\mathbf{M}_2(\mathbf{C})$. Since $[\mathbf{H} : \mathbf{R}] = 4$, $\mathbf{C}$ is a neutralizing field of $\mathbf{H}$ (Alg., chap. VIII, §10, n° 5), and the reduced norm of $q = a + b i + c j + d k \in \mathbf{H}$ is

By Alg., chap. VIII, §12, n° 3, prop. 8, one has

This stated, Prop. 17 shows that

A deeper study of the structure of locally compact fields will be made in CA, VI, §9.

(2) This also follows from Cor. 1 of Prop. 15 and the fact that left-multiplication by $z = a + i b$ has matrix $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ with respect to the canonical basis $1, i$ of $\mathbf{C}$ over $\mathbf{R}$.