It follows that, to compute in a finite field of non prime order, one needs to generate an irreducible polynomial.
12.
An extension field is constructed from an underlying prime field ( the base field ) using an irreducible polynomial over the field.
13.
In fact, this generator is a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.
14.
The fact that \ Phi _ n is an irreducible polynomial of degree \ varphi ( n ) in the Eisenstein's criterion.
15.
Addition is simply XOR . Multiplication is modulo irreducible polynomial x ^ 8 + x ^ 4 + x ^ 3 + x + 1.
16.
If, that is, one may choose as a quadratic non-residue, which allows us to have a very simple irreducible polynomial.
17.
In the preceding theorem, one may replace " distinct irreducible polynomials " by " pairwise coprime polynomials that are coprime with their derivative ".
18.
This means that it is generated by an element ? which is a root of an irreducible polynomial of degree " d ".
19.
:: : Ok, now for n = 21, we first need an irreducible polynomial of degree 21 over GF ( 2 ).
20.
The aim of factoring is usually to reduce something to basic building blocks , such as numbers to prime numbers, or polynomials to irreducible polynomials.
