Editing Cayley–Hamilton theorem (section)

===Algebraic number theory===
The Cayley–Hamilton theorem is an effective tool for computing the minimal polynomial of [[algebraic integer]]s. For example, given a finite extension <math>\mathbb{Q}[\alpha_1,\ldots,\alpha_k]</math> of <math>\mathbb{Q}</math> and an algebraic integer <math>\alpha \in \mathbb{Q}[\alpha_1,\ldots,\alpha_k]</math> which is a non-zero linear combination of the <math>\alpha_1^{n_1}\cdots\alpha_k^{n_k}</math> we can compute the minimal polynomial of <math>\alpha</math> by finding a matrix representing the <math>\mathbb{Q}</math>-[[linear transformation]]
<math display="block">\cdot \alpha : \mathbb{Q}[\alpha_1,\ldots,\alpha_k] \to \mathbb{Q}[\alpha_1,\ldots,\alpha_k]</math>
If we call this transformation matrix <math>A</math>, then we can find the minimal polynomial by applying the Cayley–Hamilton theorem to <math>A</math>.<ref>{{cite book|last1=Stein|first1=William|title=Algebraic Number Theory, a Computational Approach | page=29|url=http://wstein.org/books/ant/ant.pdf}}</ref>