Search results
Results from the WOW.Com Content Network
Irreducible polynomial. In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the ...
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 closely related necklace function N q (n) counts monic polynomials of degree n which are primary (a power of an irreducible); or alternatively irreducible polynomials of all degrees d ...
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). An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n ...
For applying the above general construction of finite fields in the case of GF(p 2), one has to find an irreducible polynomial of degree 2. For p = 2, this has been done in the preceding section. If p is an odd prime, there are always irreducible polynomials of the form X 2 − r, with r in GF(p).
Definition. Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements pi of R: x = p1p2 ⋅⋅⋅ pn with n ≥ 1.
Galois group. In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in ...
GF (2) GF (2) (also denoted , Z/2Z or ) is the finite field with two elements. [ 1 ][ a ] GF (2) is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively 0 and 1, as usual. The elements of GF (2) may be identified with the two possible values of a ...
The quotient ring is naturally isomorphic to , and is the zero ring , since, by our definition, for any , we have that , which equals itself. This fits with the rule of thumb that the larger the ideal , the smaller the quotient ring . If is a proper ideal of , i.e., , then is not the zero ring. Consider the ring of integers and the ideal of ...