Minimal polynomial

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

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 $4I_{n}$, which has characteristic polynomial $(x-4)^{n}$. However, the minimal polynomial is $x-4$, since $4I-4I=0$ as desired, so they are different for $n\geq 2$. That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.