In numerical analysis, "'Wilkinson's polynomial "'is a specific polynomial which was used by James H . Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial : the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial.
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If the coefficients do not belong to "'F " "'p ", the " p "-th root of a polynomial with zero derivative is obtained by the same substitution on " x ", completed by applying the inverse of the Frobenius automorphism to the coefficients.
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Note that you can't write the entire thing in one formula in terms of the coefficients of the polynomials, because by a result of Abel, there is no formula ( in terms of finitely many radicals ) for the roots of a polynomial .-talk [ + ] 05 : 05, 19 March 2006 ( UTC)
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Now consider the subset of real numbers that can be exactly described using some symbolic representation, such as the root of a polynomial, the sum of an infinite series, the limit to some equation, or some yet-to-be-invented notation ( think integrals before Leibniz ) . " Is there a name for such numbers ?"
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More precisely, only totally ramified primes have a chance of being Eisenstein primes for the polynomial . ( In quadratic fields, ramification is always total, so the distinction is not seen in the quadratic case like above . ) In fact, Eisenstein polynomials are directly linked to totally ramified primes, as follows : if a field extension of the rationals is generated by the root of a polynomial that is Eisenstein at then is totally ramified in the extension, and conversely if is totally ramified in a number field then the field is generated by the root of an Eisenstein polynomial at.
