Search results
Results from the WOW.Com Content Network
The case of the 105th cyclotomic polynomial is interesting because 105 is the least positive integer that is the product of three distinct odd prime numbers (3×5×7) and this polynomial is the first one that has a coefficient other than 1, 0, or −1: [3]
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
Cyclic group, a group generated by a single element; Cyclic homology, an approximation of K-theory used in non-commutative differential geometry; Cyclic module, a module generated by a single element; Cyclic notation, a way of writing permutations; Cyclic number, a number such that cyclic permutations of the digits are successive multiples of ...
A BCH code with = is called a narrow-sense BCH code.; A BCH code with = is called primitive.; The generator polynomial () of a BCH code has coefficients from (). In general, a cyclic code over () with () as the generator polynomial is called a BCH code over ().
When for all , this polynomial is irreducible in K[X], and its splitting field over K is a cyclic extension of K of degree p. This follows since for any root β , the numbers β + i , for 1 ≤ i ≤ p {\displaystyle 1\leq i\leq p} , form all the roots—by Fermat's little theorem —so the splitting field is K ( β ) {\displaystyle K(\beta )} .
A cyclic permutation consisting of a single 8-cycle. There is not widespread consensus about the precise definition of a cyclic permutation. Some authors define a permutation σ of a set X to be cyclic if "successive application would take each object of the permuted set successively through the positions of all the other objects", [1] or, equivalently, if its representation in cycle notation ...
The reciprocal of a polynomial generates the same codewords, only bit reversed — that is, if all but the first bits of a codeword under the original polynomial are taken, reversed and used as a new message, the CRC of that message under the reciprocal polynomial equals the reverse of the first bits of the original codeword. But the reciprocal ...
Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1 (equivalently, plethystic exponentials satisfy the usual properties of an exponential), and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of t m.