Minimal polynomial of an algebraic element

Definition

Let $L/K$ be a field extension and let $a \in L$ be algebraic over $K$. The minimal polynomial of $a$ over $K$ is the monic polynomial $m_a \in K[x]$ of smallest degree such that $m_a(a) = 0$.

Remarks

The minimal polynomial exists and is unique. If two monic polynomials of the smallest degree vanish at $a$, their difference has smaller degree and vanishes at $a$, so it is zero. The minimal polynomial is irreducible, and it divides each polynomial that vanishes at $a$. The minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$ is $x^2 - 2$. Over $\mathbb{R}$, it is $x - \sqrt{2}$. The minimal polynomial of $i$ over $\mathbb{R}$ is $x^2 + 1$.

Explanations

L1 human

Among all the polynomial equations that an algebraic element satisfies, the minimal polynomial is the simplest one. It depends on the base field: a bigger base field can give a smaller minimal polynomial.