## Reduced characteristic polynomials; an example

Posted: October 11, 2011 in Noncommutative Ring Theory Notes, Simple Rings
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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!).