perustekijäpolynomista

Perustekijäpolynomi, or minimal polynomial in English, is a fundamental concept in abstract algebra, particularly in the study of field extensions and linear algebra. Given a field extension K over F, and an element α in K, the perustekijäpolynomi of α over F is the unique monic polynomial of least degree with coefficients in F that has α as a root. If no such non-zero polynomial exists, then α is called transcendental over F.

The existence and uniqueness of the perustekijäpolynomi are guaranteed under certain conditions. If α is algebraic over F, meaning it is a root of some non-zero polynomial in F[x], then the set of all polynomials in F[x] that have α as a root forms an ideal in the polynomial ring F[x]. Since F[x] is a principal ideal domain, this ideal is generated by a single polynomial. This generator, when made monic, is precisely the perustekijäpolynomi.

The degree of the perustekijäpolynomi is equal to the degree of the field extension F(α) over F. Furthermore, the perustekijäpolynomi is irreducible over F. This irreducibility is a key property that links the concept to field extensions and Galois theory. In linear algebra, the concept is also related to the minimal polynomial of a linear operator on a vector space, which is the monic polynomial of least degree that annihilates the operator.