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[citation needed] [1] Irreducible polynomials over finite fields are also useful for pseudorandom number generators using feedback shift registers and discrete logarithm over F 2 n. The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces, given by Moreau's necklace-counting function M q (n).
The polynomial factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order q. This implies that, if q = p n then X q − X is the product of all monic irreducible polynomials over GF(p), whose degree divides n.
A linearised polynomial L(x) over F q is symbolically irreducible if and only if its conventional q-associate l(x) is irreducible over F q. Every q-polynomial L(x) over F q of degree > 1 has a symbolic factorization into symbolically irreducible polynomials over F q and this factorization is essentially unique (up to rearranging factors and ...
Suppose that the variables X i are given weights w i, and F(X 1, ..., X n) is a homogeneous polynomial of weight w. Then if c is coprime to w and R is a UFD and either every finitely generated projective module over R is free or c is 1 mod w, the ring R[X 1, ..., X n, Z]/(Z c − F(X 1, ..., X n)) is a UFD. [3]
An arbitrary polynomial f with coefficients in some field F is said to have distinct roots or to be square-free if it has deg f roots in some extension field.For instance, the polynomial g(X) = X 2 − 1 has precisely deg g = 2 roots in the complex plane; namely 1 and −1, and hence does have distinct roots.
A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root).
For univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this to efficiently factorize the polynomial into square-free factors, that is, factors that are not a multiple of a square, performing a sequence of GCD computations starting with gcd(f(x), f '(x)). To factorize the ...
The monic irreducible polynomial x 8 + x 4 + x 3 + x + 1 over GF(2) is not primitive. Let λ be a root of this polynomial (in the polynomial representation this would be x), that is, λ 8 + λ 4 + λ 3 + λ + 1 = 0. Now λ 51 = 1, so λ is not a primitive element of GF(2 8) and generates a multiplicative subgroup of order 51. [5]