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Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.


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01 Jan 2015
TL;DR: In this paper, the authors illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations $+,\,-, \,\cdot,\,:\,$, and $\ root n \of{\cdot}$.
Abstract: We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations $+,\,-,\,\cdot,\,:\,$, and $\root n \of{\cdot}$. Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.

3 citations

Posted Content
Rafe Jones1
TL;DR: In this article, the authors give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property, and depends on new criteria ensuring all iterates of f are irreduceible.
Abstract: We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property, and it depends on new criteria ensuring all iterates of f are irreducible. In particular when F is a number field in which the ideal (2) is not a square, we construct infinitely many families of quadratic f such that every iterate f^n is irreducible over F, but f^n is reducible modulo all primes of F for n at least 2. We also give an example for each n of a quadratic f with integer coefficients whose iterates are all irreducible over the rationals, whose (n-1)st iterate is irreducible modulo some primes, and whose nth iterate is reducible modulo all primes. From the perspective of Galois theory, this suggests that a well-known rigidity phenomenon for linear Galois representations does not exist for Galois representations obtained by polynomial iteration. Finally, we study the number of primes P for which a given quadratic f defined over a global field has f^n irreducible modulo P for all n.

3 citations

Proceedings ArticleDOI
25 Jul 2007
TL;DR: It is proved that the absolute value of every coefficient of f -- f is || f --∞ with at most one exception and the problem is reduced to solving systems of algebraic equations.
Abstract: For a real univariate polynomial f and a bounded closed domain D ⊂ C whose boundary C is a simple closed curve of finite length and is represented by a piecewise rational function, we provide a rigorous method for finding the real univariate polynomial f such that f has a zero in D and ||f -- f||∞ is minimal. First, we prove that the absolute value of every coefficient of f -- f is ||f -- f∞ with at most one exception. Using this property and the representation of C, we reduce the problem to solving systems of algebraic equations, each of which consists of two equations with two variables. Furthermore, every equation is of degree one with respect to one of the two variables.

3 citations

Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient condition that every division ring with center k and index equal to the order of G be a crossed product for G, is that k and G satisfy the hypothesis of the above theorem.

3 citations

Book ChapterDOI
Nobuo Nakagawa1
01 Jan 2002
TL;DR: In this article, it was shown that there is a relation between planar functions of elementary abelian groups and bent polynomials, and several results concerning them were proved.
Abstract: It is shown that there is a relation between planar functions of elementary abelian groups and bent polynomials. Moreover we prove several results concerning them.

3 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20234
20227
20216
202010
20196
20186