Thus, if one allows only real numbers, then irreducible polynomials are of degree either 1 or 2.

It is normally assumed that p should be an irreducible polynomial.

Over a perfect field, all irreducible polynomials are separable, so that condition is also necessary.

Three of these are Hall systems generated by the irreducible polynomials , or .

Each absolutely irreducible polynomial has infinitely many -rational points.

Irreducible polynomials allow to construct the finite fields of non prime order.

By the last condition above, if an irreducible polynomial does not have distinct roots, its derivative must be zero.

Let be a monic, irreducible polynomial of degree .

Every irreducible polynomial over k has distinct roots.

This implies that the roots of an irreducible polynomial may not be distinguished through algebraic relations.

Thus, the only non-constant irreducible polynomials over are linear polynomials.

Using this idea inductively we can write R(x) as a sum with denominators powers of irreducible polynomials.

It follows that, to compute in a finite field of non prime order, one needs to generate an irreducible polynomial.

Principal ideals generated by irreducible polynomials.

These sequences may be represented as coefficients of irreducible polynomials in a polynomial ring over Z/2Z.

On the structure of the irreducible polynomials over local fields, J. Number Theory, Vol.

Let be an irreducible polynomial of degree over , and let be the period of .

Any polynomial may be decomposed into the product of a constant by a product of irreducible polynomials.

The following five polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root.

To deal with this, the Galois field is introduced with , where for a suitable irreducible polynomial of degree .

Key-Generation Algorithm: Input: , irreducible polynomial of degree .

The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials.

Information on Primitive and Irreducible Polynomials, The (Combinatorial) Object Server.

Note that is an irreducible polynomial in the polynomial ring, and hence the code is an irreducible code.