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In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. The properties of the minimal polynomial depend on the algebraic structure to which α belongs.

Field theory

In field theory, given a field extension E / F and an element α of E which is algebraic over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p(α) = 0. The minimal polynomial is irreducible, and any other non-zero polynomial f with f(α) = 0 is a multiple of p.

For example, for ${\displaystyle F=\mathbb {Q} ,E=\mathbb {R} ,\alpha ={\sqrt {2}}}$ the minimal polynomial for ${\displaystyle \alpha }$ is ${\displaystyle p(x)=x^{2}-2}$. If ${\displaystyle \alpha ={\sqrt {2}}+{\sqrt {3}}}$ then ${\displaystyle p(x)=x^{4}-10x+1=(x-{\sqrt {2}}-{\sqrt {3}})(x+{\sqrt {2}}-{\sqrt {3}})(x-{\sqrt {2}}+{\sqrt {3}})(x+{\sqrt {2}}+{\sqrt {3}})}$ is the minimal polynomial.

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

Linear algebra

In linear algebra, the minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F of least degree such that p(A)=0. Any other polynomial q with q(A) = 0 is a (polynomial) multiple of p.

The following three statements are equivalent:

1. λ∈F is a root of p(x),
2. λ is a root of the characteristic polynomial of A,
3. λ is an eigenvalue of A.

The multiplicity of a root λ of p(x) is the size of the largest Jordan block corresponding to λ.

The minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix ${\displaystyle 4I_{n}}$, which has characteristic polynomial ${\displaystyle (x-4)^{n}}$. However, the minimal polynomial is ${\displaystyle x-4}$, since ${\displaystyle 4I-4I=0}$ as desired, so they are different for ${\displaystyle n\geq 2}$. That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.

• Linear algebra
• Polynomials
• Field theory