Let p ( x ) be an irreducible polynomial of degree m over \ mathbb { F } _ 2, and let p be the period of p ( x ).
32.
Irreducible polynomials over finite fields are also useful for Pseudorandom number generators using feedback shift registers and discrete logarithm over "'F "'2 " n ".
33.
When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials ( see Factorization of polynomials ).
34.
This means that every element of these rings is a product of a constant and a product of irreducible polynomials ( those that are not the product of two non-constant polynomials ).
35.
In fact, if is a prime number, and, the field of order is a simple algebraic extension of the prime field of elements, generated by a root of an irreducible polynomial of degree.
36.
Every irreducible closed subset of "'P " "'n " of codimension one is a hypersurface; i . e ., the zero set of some homogeneous irreducible polynomial.
37.
Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing " P ", but leaving " L " the same.
38.
It follows that the th cyclotomic polynomial factors over into distinct irreducible polynomials that have all the same degree, say, and that is the smallest field of characteristic that contains the th primitive roots of unity.
39.
The minimal polynomial of an algebraic element " x " of " L " is irreducible, and is the unique monic irreducible polynomial of which " x " is a root.
40.
For example, let " P " be an irreducible polynomial with integer coefficients and " p " be a prime number which does not divide the leading coefficient of " P ".
