Трансформация полиномдардың түбірлері арқылы жүзеге асуы

Transformation of a polynomial induced by a transformation of its roots

In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.

If the polynomial P is irreducible, then either the resulting polynomial Q is irreducible, or it is a power of an irreducible polynomial. Let be a root of P and consider L, the field extension generated by The former case means that is a primitive element of L, which has Q as minimal polynomial. In the latter case, belongs to a subfield of L and its minimal polynomial is the irreducible polynomial that has Q as power.

Polynomial transformations have been applied to the simplification of polynomial equations for solution, where possible, by radicals. Descartes introduced the transformation of a polynomial of degree d which eliminates the term of degree d − 1 by a translation of the roots. Such a polynomial is termed depressed. This already suffices to solve the quadratic by square roots. In the case of the cubic, Tschirnhaus transformations replace the variable by a quadratic function, thereby making it possible to eliminate two terms, and so can be used to eliminate the linear term in a depressed cubic to achieve the solution of the cubic by a combination of square and cube roots. The Bring–Jerrard transformation, which is quartic in the variable, brings a quintic into Bring Jerrard normal form with terms of degree 5,1, and 0.