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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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. Field theory is a branch of mathematics which studies the properties of fields. ... In abstract algebra, a subfield of a field L is a subset K of L which is closed under the addition and multiplication operations of L and itself forms a field with these operations. ... In mathematics, the roots of polynomials are in abstract algebra called algebraic elements. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ...

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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... This article presents the essential definitions. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K, to what extent M can be simplified into a standard...

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 . That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem. In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation. ...