polynomials and minimality

Question

Could someone explain the concept of minimal polynomials? It seems like these are polynomials which cant be reduced further, but at the same time I am confused cause when we consider $\mathbb Z_2[x]$ why is $1+x+x^4$ a minimal polynomial of $x^{15}-1$ and not $1+x^3+x^4$ or $1+x+x^2$. I am not sure if my understanding is wrong. Could someone clarify!

Answer 1

As suggested in the comments, minimal polynomials don't come on their own. Every minimal polynomial is the minimal polynomial of some object. In the basic examples, this object can be an element in a field (in the context of field extensions), or a square matrix.

Matrix: Let $A$ be an $n\times n$ matrix whose entries are in the field $\mathbb{F}$. Given a polynomial $p\in\mathbb{F}[x]$, one can substitute $A$ in $p$ and obtain another matrix, $p(A)$. The minimal polynomial of A, which is denoted by $m_A$, is a monic polynomial that satisfies (a) $m_A(A)=0$ and (b) no polynomial of degree smaller than that of $m_A$ satisfies (a). For example, if $A=I$, the identity matrix, then the minimal polynomial of $A$ is $m_A(x)=x-1.$

In the context of field extensions: If $\mathbb{F}\subset\mathbb{K}$ is a field extension, and $\alpha\in\mathbb{K}$, then $\alpha$ can be substituted in any polynomial $p\in\mathbb{F}[x]$. The minimal polynomial of $\alpha$ over $\mathbb{F}$, which is denoted by $m_\alpha$, is a monic polynomial in $\mathbb{F}[x]$, that satisfies (a) $m_\alpha(\alpha)=0$ and (b) no polynomial of smaller degree (with coefficients in $\mathbb{F}$!) satisfies (a). For example, the minimal polynomial of $i$ over $\mathbb{R}$ is $m_i(x)=x^2+1$, and the minimal polynomial of $i$ over $\mathbb{C}$ is $m_\alpha=x-i$.

Note that the minimal polynomial of $\alpha$ over $\mathbb{F}$ is always irreducible (why?), whereas the minimal polynomial of a matrix need not to be irreducible (can you think of an example showing that?).