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Every polynomial over a field F may be factored into a product of a non-zero constant and a finite number of irreducible (over F) polynomials. This decomposition is unique up to the order of the factors and the multiplication of the factors by non-zero constants whose product is 1.
The polynomial P = x 4 + 1 is irreducible over Q but not over any finite field. On any field extension of F 2 , P = ( x + 1) 4 . On every other finite field, at least one of −1, 2 and −2 is a square, because the product of two non-squares is a square and so we have
If F is a finite field, a non-constant monic polynomial with coefficients in F is irreducible over F, if it is not the product of two non-constant monic polynomials, with coefficients in F. As every polynomial ring over a field is a unique factorization domain , every monic polynomial over a finite field may be factored in a unique way (up to ...
A monic irreducible polynomial of degree n having coefficients in the finite field GF(q), where q = p t for some prime p and positive integer t, is called a primitive polynomial if all of its roots are primitive elements of GF(q n). [2] [3] In the polynomial representation of the finite field, this implies that x is a primitive element.
Similarly, an irreducible module is another name for a simple module. Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as ...
In mathematics, the Conway polynomial C p,n for the finite field F p n is a particular irreducible polynomial of degree n over F p that can be used to define a standard representation of F p n as a splitting field of C p,n. Conway polynomials were named after John H. Conway by Richard A. Parker, who was the first to define them and compute ...
The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of () into powers of irreducible polynomials (recalling that the ring of polynomials over a finite field is a unique factorization domain).
A field F is perfect if and only if all irreducible polynomials are separable. It follows that F is perfect if and only if either F has characteristic zero, or F has (non-zero) prime characteristic p and the Frobenius endomorphism of F is an automorphism. This includes every finite field.