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TRINITY COLLEGE 2006 Course 4281 Prime Numbers Bernhard
TRINITY COLLEGE 2006 Course 4281 Prime Numbers Bernhard

PDF of Version 2.0-T of GIAA here.
PDF of Version 2.0-T of GIAA here.

When an Extension of Nagata Rings Has Only Finitely Many
When an Extension of Nagata Rings Has Only Finitely Many

An Introduction to Algebraic Number Theory, and the Class Number
An Introduction to Algebraic Number Theory, and the Class Number

... We describe various algebraic invariants of number fields, as well as their applications. These applications relate to prime ramification, the finiteness of the class number, cyclotomic extensions, and the unit theorem. Finally, we present an exposition of the class number formula, which generalizes ...
1 Divisibility. Gcd. Euclidean algorithm.
1 Divisibility. Gcd. Euclidean algorithm.

Fibonacci Integers - Dartmouth Math Home
Fibonacci Integers - Dartmouth Math Home

Asymptotic formulæ for the distribution of integers of various types∗
Asymptotic formulæ for the distribution of integers of various types∗

Regularization Tools
Regularization Tools

Using extended feature objects for partial similarity
Using extended feature objects for partial similarity

VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert
VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert

HERE - University of Georgia
HERE - University of Georgia

1 - University of Notre Dame
1 - University of Notre Dame

NON-SPLIT REDUCTIVE GROUPS OVER Z Brian
NON-SPLIT REDUCTIVE GROUPS OVER Z Brian

MONOMIAL IDEALS, ALMOST COMPLETE INTERSECTIONS AND
MONOMIAL IDEALS, ALMOST COMPLETE INTERSECTIONS AND

ON THE FIELDS GENERATED BY THE LENGTHS OF CLOSED
ON THE FIELDS GENERATED BY THE LENGTHS OF CLOSED

Garrett 09-23-2011 1 Continuing the pre/review of the simple (!?) case... Some
Garrett 09-23-2011 1 Continuing the pre/review of the simple (!?) case... Some

... Riemann’s explicit formula connects complex zeros of meromorphic continuations of zeta functions (and L-functions) to tangible, finitistic properties of primes. Gauss’ Quadratic Reciprocity is proven via Gauss sums, which are Lagrange resolvents for cyclotomic fields. Dedekind zeta functions of quad ...
Math 780: Elementary Number Theory
Math 780: Elementary Number Theory

lecture notes - TU Darmstadt/Mathematik
lecture notes - TU Darmstadt/Mathematik

pdf
pdf

day five- inverses and logarithms
day five- inverses and logarithms

Sylow`s Subgroup Theorem
Sylow`s Subgroup Theorem

Explicit Estimates in the Theory of Prime Numbers
Explicit Estimates in the Theory of Prime Numbers

Factor This - Yeah, math, whatever.
Factor This - Yeah, math, whatever.

... (a) x 2  8 x  7. (b) x 2  2 x  15 . (c) x 2  x  20. (d) x 2  8 x  12. (e) 7 x 2  14 xy  21y . (6) Factor the polynomial: (a) 6x2 – x – 15 (b) 5x2 – 7x – 6 (c) 8x2 + 22x – 21 (7) Factor the polynomial: (a) x3  1000 (b) 27 x3  64 (c) 5 x 3  5 (8) Factor the polynomial: (a) x 2  10 x  25 ...
ON SEQUENCES DEFINED BY LINEAR RECURRENCE
ON SEQUENCES DEFINED BY LINEAR RECURRENCE

Topological realizations of absolute Galois groups
Topological realizations of absolute Galois groups

< 1 2 3 4 5 6 7 8 9 ... 231 >

Factorization of polynomials over finite fields

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.The case of the factorization of univariate polynomials over a finite field, which is the subject of this article, is especially important, because all the algorithms (including the case of multivariate polynomials over the rational numbers), which are sufficiently efficient to be implemented, reduce the problem to this case (see Polynomial factorization). It is also interesting for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.
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