Minimal polynomial

The minimal polynomial of an algebraic element over a field is the unique monic irreducible polynomial over that field having the element as a root.

Field extension: A field extension is a field containing a specified smaller field as a subfield. The base field and algebraic element sit inside this larger field structure.
Evaluation homomorphism: An evaluation homomorphism maps a polynomial to its value at a specified element. Its kernel consists exactly of the polynomials that vanish at the element.
Degree of a field extension: The degree of a field extension is its dimension as a vector space over the smaller field. For an algebraic element, this degree equals the degree of its minimal polynomial.
Characteristic polynomial: The characteristic polynomial of a linear operator is the determinant of the variable times the identity minus its matrix. It shares roots and divisibility relations with the minimal polynomial but can have larger degree.
Polynomial ring: A polynomial ring consists of polynomials with coefficients in a given ring, with usual addition and multiplication. The minimal polynomial is an element of the polynomial ring over its base field.
Kernel of a ring homomorphism: The kernel of a ring homomorphism is the set of elements mapped to zero. The minimal polynomial generates the evaluation kernel for an algebraic element over a field.
Power basis: A power basis is a basis formed by consecutive powers of one element in a field extension. Powers below the minimal polynomial's degree form a basis for the simple extension.
Annihilating polynomial: An annihilating polynomial is any polynomial that evaluates to zero at a specified algebraic element or operator. The minimal polynomial is the monic irreducible generator among all annihilating polynomials.
Irreducible polynomial: An irreducible polynomial over a field has positive degree and cannot be factored into lower-degree polynomials over that field. Irreducibility distinguishes the minimal polynomial from other polynomials that vanish at the element.
Polynomial division: Polynomial division expresses one polynomial as a quotient times a nonzero divisor plus a remainder of smaller degree. Division shows that every vanishing polynomial is divisible by the minimal polynomial.

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