**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

*a*

_{i}are independently and identically distributed with a mean of zero, the real roots are mostly located near ±1. The complex roots can be shown to be on or close to the unit circle.

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

^{}

^{}

*σ*, a similar but more complex formula is known .

^{ }

## No comments:

## Post a Comment