Reduced characteristic polynomials; an example

Posted: October 11, 2011 in Noncommutative Ring Theory Notes, Simple Rings
Tags: , ,

Recall that \mathbb{H}, the real quaternion algebra, as a vector space over \mathbb{R} has a basis \{1,i,j,k \} and multiplication of two elements in \mathbb{H} is done using distributive law and the relations i^2=j^2=k^2=ijk=-1. It is easy to see that \mathbb{H} is a division algebra and its center is \mathbb{R}. So, by the lemma in this post, \mathbb{H} \otimes_{\mathbb{R}} \mathbb{C} \cong M_2(\mathbb{C}). This isomorphism is also a result of this fact that \mathbb{C} is a maximal subfield of \mathbb{H} (see corollary 3 in this post!). In the following example, we are going to find the reduced characteristic polynomial of an element of \mathbb{H} by giving a \mathbb{C}-algebra isomorphism \mathbb{H} \otimes_\mathbb{R} \mathbb{C} \longrightarrow M_2(\mathbb{C}) explicitely. We will learn later how to find the reduced characteristic polynomial of an element of any finite dimensional central division algebra.

Example. If a = \alpha + \beta i + \gamma j + \delta k \in \mathbb{H}, then \text{Prd}_{\mathbb{H}}(a,x)=x^2 - 2 \alpha x + \alpha^2+\beta^2+\gamma^2 + \delta^2.

Solution. We define f : \mathbb{H} \otimes_{\mathbb{R}} \mathbb{C} \longrightarrow M_2(\mathbb{C}) as follows: for every a_1 = \alpha_1 + \beta_1 i + \gamma_1 j + \delta_1 k \in \mathbb{H} and z \in \mathbb{C} we define

f(a_1 \otimes_{\mathbb{R}} z) = \begin{pmatrix} (\alpha_1 + \beta_1 i)z & (\gamma_1 + \delta_1 i)z \\ (-\gamma_1 + \delta_1 i)z & (\alpha_1 - \beta_1 i)z \end{pmatrix}.

Then of course we extend f linearly to all elements of \mathbb{H} \otimes_{\mathbb{R}} \mathbb{C}. See that f is a \mathbb{C}-algebra homomorphism. Since \mathbb{H} \otimes_{\mathbb{R}} \mathbb{C} is simple, f is injective. Therefore f is an isomorphism because 4=\dim_{\mathbb{C}} M_2(\mathbb{C}) = \dim_{\mathbb{C}} \mathbb{H} \otimes_{\mathbb{R}} \mathbb{C}. It is easy now to find \text{Prd}_{\mathbb{H}}(a):

\text{Prd}_{\mathbb{H}}(a,x)=\det(xI - f(a \otimes_{\mathbb{R}} 1)) = \det \left ( \begin{pmatrix} x & 0 \\ 0 & x \end{pmatrix} - \begin{pmatrix} \alpha + \beta i & \gamma + \delta i \\ -\gamma + \delta i & \alpha - \beta i \end{pmatrix} \right).

The rest of the proof is straightforward. \Box

In the above example, the coefficients of \text{Prd}_{\mathbb{H}}(a,x) are all in \mathbb{R}. This is not an accident. We have already proved that the reduced characteristic polynomial of an element of a finite dimensional central simple k-algebra is always in k[x] (see the theorem in this post!).

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