# What is the relation between minimal polynomial of a matrix and of an algebraic integer?

Both matrices and Algebraic integers have minimal polynomials. But I struggle to get how these two types of minimal polynomials are related. Well, they are two different rings, but I think we can find some connection between them

For example, if an algebraic number $$\alpha$$ can be expressed as a combination of the linear independent basis $$1,\sqrt{3}, \sqrt{5},\sqrt{15}$$, then $$\alpha^n$$ can be expressed as a combination of this basis as well. The equation $$\sum_{i=0}^4 c_i\alpha^i=0$$ is essentially four linear equations with four unknowns $$c_i$$. So, in general, the values of $$c_i$$ are uniquely determined, and we therefore find the minimal polynomial.

The process above is essentially linear algebra. However, I find that the power $$\alpha^n$$ cannot be easily represented by the power of a matrix. Is there an easy way to embed the algebraic integers into a set of matrices so that the minimal polynomial of algebraic integers can be found easily by finding the minimal polynomial of matrices?

• If you know the minimal polynomial $p(x)$ then you can construct a companion matrix which has characteristic polynomial equal to $p(x)$, but not quite what you are looking for. On the other hand there are formulas for the coefficients of $p(x)$ in terms of conjugates of $\alpha$, so there is some relationship I guess? Feb 22, 2019 at 12:17

Let $$\alpha$$ be an algebraic integer, let $$K={\bf Q}(\alpha)$$. Then multiplication by $$\alpha$$ is a $$\bf Q$$-linear transformation on $$K$$. Let $$B$$ be a basis for $$K$$ as a $$\bf Q$$-vector space, and let $$M$$ be the matrix representing multiplication by $$\alpha$$ with respect to $$B$$. Then the minimal polynomial for $$\alpha$$ over $$\bf Q$$ is the minimal polynomial for $$M$$.
Let the degree of $$\alpha$$ over $$\bf Q$$ be $$n$$. Then you can choose the basis $$\{\,1,\alpha,\alpha^2,\dots,\alpha^{n-1}\,\}$$, and the matrix $$M$$ that you get has a particularly simple form; it's called the companion matrix for $$\alpha$$. But it requires that you know how to express $$\alpha^n$$ in terms of the basis, which is the same as knowing the minimal polynomial of $$\alpha$$.
• In the general case (if $K\neq Q(a)$), the characterisic polynomial of $a$ is a power of its minimal polynomial. See e.g. math.stackexchange.com/a/2710883/300700 Feb 23, 2019 at 6:59