Definition :- Let F be any field and p(x) be any polynomial in F[x] . Then an element a lying in some extension field of F is called a root of p(x) if p(a) = 0 .
from the theorems of polynomials we know that the element lying in an extension K of F which were algebraic over F i.e. which satisfies a polynomial
p(x) is element of F[x] i.e. p(a) = 0
Here in this article we shall aim at finding an extension field K of F in which a given polynomial
p(x) is an element of F[x] has a root .
The statistical properties of the roots of a random polynomial have been the subject of several studies. Let
If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula