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Here is my current understanding of what a minimal polynomial is:

$p(x)$ is the minimal polynomial of $a$ over a field $\mathbb{F}$ if $p(a) = 0$ and $p(x)$ is the lowest degree irreducible, monic polynomial in $\mathbb{F}[x]$ for which this is true.

Is this exactly correct? Please tell me if not.

Now my question is, what are minimal polynomials used for? What applications does it have in algebra research? Does it tell us anything about the field we are working in?

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    $\begingroup$ Here is a "funny application" of minimal polynomials: you can prove that every real number which can be constructed using only a straightedge and a compass is algebraic over $\Bbb Q$, and its minimal polynomial has a degree $2^n$ (for some $n \geq 0$) [NB : the converse doesn't hold]. But the minimal polynomial of $\sqrt[3]{2}$ over $\Bbb Q$ has degree $3$, so that $\sqrt[3]{2}$ is not a constructible number. Therefore "doubling the cube" is impossible (using only unruled straightedge and compass). $\endgroup$
    – Watson
    Commented Dec 22, 2016 at 16:50
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    $\begingroup$ More conceptually it is the gcd of the set of all polynomials $\in \Bbb F[x]\,$ having $a$ as a root (or the generator of the associated ideal if that is familiar). In particular $\ f(a) = 0\iff p\mid f.\ \ $ $\endgroup$ Commented Dec 22, 2016 at 18:35

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  1. For a correct definition of minimal polynomial within field theory see here. So we start with a field extension $L/K$. You did not specify where $a$ is from, and did not mention a field extension.

  2. It is widely used in field theory, but also in linear algebra (for a specific definition see here). For many proofs in abstract algebra it is quite useful. It is furthermore used for many results in algebraic number theory (involving for example norm and trace of field extensions, rings of integers, and discriminants).

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