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To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add the equation of the generator to the equations of the system. Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers.
As an example, in the polynomial ring k [X,Y] consider the ideal generated by the irreducible polynomial Y 2 − X 3 and form the field of fractions of the quotient ring k [X,Y]/(Y 2 − X 3). This is a function field of one variable over k; it can also be written as () (with degree 2 over ()) or as () (with degree 3 over ()). We see that the ...
The coefficient of the highest-degree term in the polynomial is required to be 1. More formally, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E/F. The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F.
For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and ...
A polynomial function is one that has the form = + + + + + where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on.
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. [1] Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ...
In algebra, a polynomial functor is an endofunctor on the category of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers and the exterior powers are polynomial functors from to ; these two are also Schur functors.
One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field.These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots.