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.

Let ${\displaystyle E/F}$ be a field extension, ${\displaystyle \alpha }$ an element of ${\displaystyle E}$, and ${\displaystyle F[x]}$ the ring of polynomials in ${\displaystyle x}$ over ${\displaystyle F}$. The element ${\displaystyle \alpha }$ has a minimal polynomial when ${\displaystyle \alpha }$ is algebraic over ${\displaystyle F}$, that is, when ${\displaystyle f(\alpha )=0}$ for some non-zero polynomial ${\displaystyle f(x)}$ in ${\displaystyle F[x]}$. Then the minimal polynomial of ${\displaystyle \alpha }$ is defined as the monic polynomial of least degree among all polynomials in ${\displaystyle F[x]}$ having ${\displaystyle \alpha }$ as a root.

Throughout this section, let ${\displaystyle E/F}$ be a field extension over ${\displaystyle F}$ as above, let ${\displaystyle \alpha \in E}$ be an algebraic element over ${\displaystyle F}$ and let ${\displaystyle J_{\alpha }}$ be the ideal of polynomials vanishing on ${\displaystyle \alpha }$.

The minimal polynomial ${\displaystyle f}$ of ${\displaystyle \alpha }$ is unique.

To prove this, suppose that ${\displaystyle f}$ and ${\displaystyle g}$ are monic polynomials in ${\displaystyle J_{\alpha }}$ of minimal degree ${\displaystyle n>0}$. We have that ${\displaystyle r:=f-g\in J_{\alpha }}$ (because the latter is closed under addition/subtraction) and that ${\displaystyle m:=\deg(r)<n}$ (because the polynomials are monic of the same degree). If ${\displaystyle r}$ is not zero, then ${\displaystyle r/c_{m}}$ (writing ${\displaystyle c_{m}\in F}$ for the non-zero coefficient of highest degree in ${\displaystyle r}$) is a monic polynomial of degree ${\displaystyle m<n}$ such that ${\displaystyle r/c_{m}\in J_{\alpha }}$ (because the latter is closed under multiplication/division by non-zero elements of ${\displaystyle F}$), which contradicts our original assumption of minimality for ${\displaystyle n}$. We conclude that ${\displaystyle 0=r=f-g}$, i.e. that ${\displaystyle f=g}$.

The minimal polynomial ${\displaystyle f}$ of ${\displaystyle \alpha }$ is irreducible, i.e. it cannot be factorized as ${\displaystyle f=gh}$ for two polynomials ${\displaystyle g}$ and ${\displaystyle h}$ of strictly lower degree.

To prove this, first observe that any factorization ${\displaystyle f=gh}$ implies that either ${\displaystyle g(\alpha )=0}$ or ${\displaystyle h(\alpha )=0}$, because ${\displaystyle f(\alpha )=0}$ and ${\displaystyle F}$ is a field (hence also an integral domain). Choosing both ${\displaystyle g}$ and ${\displaystyle h}$ to be of degree strictly lower than ${\displaystyle f}$ would then contradict the minimality requirement on ${\displaystyle f}$, so ${\displaystyle f}$ must be irreducible.

The minimal polynomial ${\displaystyle f}$ of ${\displaystyle \alpha }$ generates the ideal ${\displaystyle J_{\alpha }}$, i.e. every ${\displaystyle g}$ in ${\displaystyle J_{\alpha }}$ can be factorized as ${\displaystyle g=fh}$ for some ${\displaystyle h}$ in ${\displaystyle F[x]}$.

To prove this, it suffices to observe that ${\displaystyle F[x]}$ is a principal ideal domain, because ${\displaystyle F}$ is a field: this means that every ideal ${\displaystyle I}$ in ${\displaystyle F[x]}$, ${\displaystyle J_{\alpha }}$ amongst them, is generated by a single element ${\displaystyle f}$. With the exception of the zero ideal ${\displaystyle I=\{0\}}$, the generator ${\displaystyle f}$ must be non-zero and it must be the unique polynomial of minimal degree, up to a factor in ${\displaystyle F}$ (because the degree of ${\displaystyle fg}$ is strictly larger than that of ${\displaystyle f}$ whenever ${\displaystyle g}$ is of degree greater than zero). In particular, there is a unique monic generator ${\displaystyle f}$, and all generators must be irreducible. When ${\displaystyle I}$ is chosen to be ${\displaystyle J_{\alpha }}$, for ${\displaystyle \alpha }$ algebraic over ${\displaystyle F}$, then the monic generator ${\displaystyle f}$ is the minimal polynomial of ${\displaystyle \alpha }$.

Given a Galois field extension ${\displaystyle L/K}$ the minimal polynomial of any ${\displaystyle \alpha \in L}$ not in ${\displaystyle K}$ can be computed as $${\displaystyle f(x)=\prod _{\sigma \in {\text{Gal}}(L/K)}(x-\sigma (\alpha ))}$$ if ${\displaystyle \alpha }$ has no stabilizers in the Galois action. Since it is irreducible, which can be deduced by looking at the roots of ${\displaystyle f'}$, it is the minimal polynomial. Note that the same kind of formula can be found by replacing ${\displaystyle G={\text{Gal}}(L/K)}$ with ${\displaystyle G/N}$ where ${\displaystyle N={\text{Stab}}(\alpha )}$ is the stabilizer group of ${\displaystyle \alpha }$. For example, if ${\displaystyle \alpha \in K}$ then its stabilizer is ${\displaystyle G}$, hence ${\displaystyle (x-\alpha )}$ is its minimal polynomial.

If ${\displaystyle F=\mathbb {Q} }$, ${\displaystyle E=\mathbb {R} }$, ${\displaystyle \alpha ={\sqrt {2}}}$, then the minimal polynomial for ${\displaystyle \alpha }$ is ${\displaystyle a(x)=x^{2}-2}$. The base field ${\displaystyle F}$ is important as it determines the possibilities for the coefficients of ${\displaystyle a(x)}$. For instance, if we take ${\displaystyle F=\mathbb {R} }$, then the minimal polynomial for ${\displaystyle \alpha ={\sqrt {2}}}$ is ${\displaystyle a(x)=x-{\sqrt {2}}}$.

In general, for the quadratic extension given by a square-free ${\displaystyle d}$, computing the minimal polynomial of an element ${\displaystyle a+b{\sqrt {d\,}}}$ can be found using Galois theory. Then $${\displaystyle {\begin{aligned}f(x)&=(x-(a+b{\sqrt {d\,}}))(x-(a-b{\sqrt {d\,}}))\\&=x^{2}-2ax+(a^{2}-b^{2}d)\end{aligned}}}$$ in particular, this implies ${\displaystyle 2a\in \mathbb {Z} }$ and ${\displaystyle a^{2}-b^{2}d\in \mathbb {Z} }$. This can be used to determine ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d\,}}\!\!\!\;\;)}}$ through a series of relations using modular arithmetic.

If ${\displaystyle \alpha ={\sqrt {2}}+{\sqrt {3}}}$, then the minimal polynomial in ${\displaystyle \mathbb {Q} [x]}$ is ${\displaystyle a(x)=x^{4}-10x^{2}+1=(x-{\sqrt {2}}-{\sqrt {3}})(x+{\sqrt {2}}-{\sqrt {3}})(x-{\sqrt {2}}+{\sqrt {3}})(x+{\sqrt {2}}+{\sqrt {3}})}$.

Notice if ${\displaystyle \alpha ={\sqrt {2}}}$ then the Galois action on ${\displaystyle {\sqrt {3}}}$ stabilizes ${\displaystyle \alpha }$. Hence the minimal polynomial can be found using the quotient group ${\displaystyle {\text{Gal}}(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )/{\text{Gal}}(\mathbb {Q} ({\sqrt {3}})/\mathbb {Q} )}$.

The minimal polynomials in ${\displaystyle \mathbb {Q} [x]}$ of roots of unity are the cyclotomic polynomials. The roots of the minimal polynomial of 2cos(2π/n) are twice the real part of the primitive roots of unity.

The minimal polynomial in ${\displaystyle \mathbb {Q} [x]}$ of the sum of the square roots of the first ${\displaystyle n}$ prime numbers is constructed analogously, and is called a Swinnerton-Dyer polynomial.

• Ring of integers
• Algebraic number field
• Minimal polynomial (linear algebra)

• Weisstein, Eric W. "Algebraic Number Minimal Polynomial". MathWorld.
• Minimal polynomial at PlanetMath.
• Pinter, Charles C. A Book of Abstract Algebra. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270–273. ISBN 978-0-486-47417-5