Minimal polynomial (field theory)

In field theory, a branch of mathematics, the minimal polynomial of an element ${\displaystyle \alpha }$ of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that ${\displaystyle \alpha }$ is a root of the polynomial. If the minimal polynomial of ${\displaystyle \alpha }$ exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1.

More formally, a minimal polynomial is defined relative to a field extension ${\displaystyle E/F}$ and an element of the extension field ${\displaystyle E/F}$. The minimal polynomial of an element, if it exists, is a member of ${\displaystyle F[x]}$, the ring of polynomials in the variable ${\displaystyle x}$ with coefficients in ${\displaystyle F}$. Given an element ${\displaystyle \alpha }$ of ${\displaystyle E}$, let ${\displaystyle J_{\alpha }}$ be the set of all polynomials ${\displaystyle f(x)}$ in ${\displaystyle F[x]}$ such that ${\displaystyle f(\alpha )=0}$. The element ${\displaystyle \alpha }$ is called a root or zero of each polynomial in ${\displaystyle J_{\alpha }}$.

More specifically, ${\displaystyle J_{\alpha }}$ is the kernel of the ring homomorphism from ${\displaystyle F[x]}$ to ${\displaystyle E}$ which sends polynomials ${\displaystyle g}$ to their value ${\displaystyle g(\alpha )}$ at the element ${\displaystyle \alpha }$. Because it is the kernel of a ring homomorphism, ${\displaystyle J_{\alpha }}$ is an ideal of the polynomial ring ${\displaystyle F[x]}$: it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of ${\displaystyle F}$ (which is scalar multiplication if ${\displaystyle F[x]}$ is regarded as a vector space over ${\displaystyle F}$).

The zero polynomial, all of whose coefficients are 0, is in every ${\displaystyle J_{\alpha }}$ since ${\displaystyle 0\alpha ^{i}=0}$ for all ${\displaystyle \alpha }$ and ${\displaystyle i}$. This makes the zero polynomial useless for classifying different values of ${\displaystyle \alpha }$ into types, so it is excepted. If there are any non-zero polynomials in ${\displaystyle J_{\alpha }}$, i.e. if the latter is not the zero ideal, then ${\displaystyle \alpha }$ is called an algebraic element over ${\displaystyle F}$, and there exists a monic polynomial of least degree in ${\displaystyle J_{\alpha }}$. This is the minimal polynomial of ${\displaystyle \alpha }$ with respect to ${\displaystyle E/F}$. It is unique and irreducible over ${\displaystyle F}$. If the zero polynomial is the only member of ${\displaystyle J_{\alpha }}$, then ${\displaystyle \alpha }$ is called a transcendental element over ${\displaystyle F}$ and has no minimal polynomial with respect to ${\displaystyle E/F}$.

Minimal polynomials are useful for constructing and analyzing field extensions. When ${\displaystyle \alpha }$ is algebraic with minimal polynomial ${\displaystyle f(x)}$, the smallest field that contains both ${\displaystyle F}$ and ${\displaystyle \alpha }$ is isomorphic to the quotient ring ${\displaystyle F[x]/\langle f(x)\rangle }$, where ${\displaystyle \langle f(x)\rangle }$ is the ideal of ${\displaystyle F[x]}$ generated by ${\displaystyle f(x)}$. Minimal polynomials are also used to define conjugate elements.