منابع مشابه
Roots and Irreducible Polynomials
This handout, which accompanies the course on analogies between Z and F [T ], discusses some properties of polynomials in F [T ]. The results in Sections 1 and 2 work with any F , but the results in Section 3 and 4 are (somewhat) special to the field F = Fp. The main result in these notes in Theorem 3.7. The notation F [T ]h(T ), for the ring of polynomials in F [T ] considered modulo h(T ), is...
متن کاملPrime Numbers and Irreducible Polynomials
The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry. There are certain conjectures indicating that the connection goes well beyond analogy. For example, there is a famous conjecture of Buniakowski formulated in 1854 (see Lang [3, p. 323]), independently reformulated by Schinzel, to the effect that ...
متن کاملIrreducible Polynomials over Finite Fields
As we will see, modular arithmetic aids in testing the irreducibility of polynomials and even in completely factoring polynomials in Z[x]. If we expect a polynomial f(x) is irreducible, for example, it is not unreasonable to try to find a prime p such that f(x) is irreducible modulo p. If we can find such a prime p and p does not divide the leading coefficient of f(x), then f(x) is irreducible ...
متن کاملInvariants of Defectless Irreducible Polynomials
Defectless irreducible polynomials over a valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case where a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are ...
متن کاملIrreducible Polynomials of Maximum Weight
We establish some necessary conditions for the existence of irreducible polynomials of degree n and weight n over F2. Such polynomials can be used to efficiently implement multiplication in F2n . We also provide a simple proof of a result of Bluher concerning the reducibility of a certain family of polynomials.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Elemente der Mathematik
سال: 2001
ISSN: 0013-6018
DOI: 10.1007/pl00000553