MAGMA Computational Algebra System

Polynomial Factorization

Polynomials over local fields can be factored.

Factorization(f) : RngUPolElt[RngLocA] -> SeqEnum, RngElt, Any

    Certificates: BoolElt               Default: false

The factorization of the polynomial f over a local field defined by an arbitrary polynomial. The factorization is returned as a sequence of tuples of prime polynomials and exponents along with a scalar factor. If the parameter Certificates is true then certificates proving the primality of each prime are also returned.

Example RngLocA_poly-fact (H48E8)

> Q3:=pAdicField(3,40);
> Q3X<x>:=PolynomialRing(Q3);
> L<a>:=LocalField(Q3,x^6-6*x^4+9*x^2-27);
> Factorization(Polynomial(L,x^6-6*x^4+9*x^2-27));
[
    <(O(3^38)*a^5 + O(3^38)*a^4 + O(3^39)*a^3 + O(3^39)*a^2 + O(3^40)*a + (1 + 
        O(3^40)))*$.1 + (5*3^35 + O(3^38))*a^5 + O(3^38)*a^4 - (10*3^36 + 
        O(3^39))*a^3 + O(3^39)*a^2 + (2701703435345984179 + O(3^40))*a + 
        O(3^40), 1>,
    <(O(3^38)*a^5 + O(3^38)*a^4 + O(3^39)*a^3 + O(3^39)*a^2 + O(3^40)*a + (1 + 
        O(3^40)))*$.1 + -(29642867960*3^-1 + O(3^23))*a^5 + O(3^24)*a^4 + 
        (148214339800*3^-1 + O(3^24))*a^3 + O(3^25)*a^2 - (116512378274*3 + 
        O(3^25))*a + O(3^26), 1>,
    <(O(3^38)*a^5 + O(3^38)*a^4 + O(3^39)*a^3 + O(3^39)*a^2 + O(3^40)*a + (1 + 
        O(3^40)))*$.1 + -(29642867960*3^-1 + O(3^23))*a^5 + O(3^24)*a^4 + 
        (148214339800*3^-1 + O(3^24))*a^3 + O(3^25)*a^2 - (349537134821 + 
        O(3^25))*a + O(3^26), 1>,
    <(O(3^38)*a^5 + O(3^38)*a^4 + O(3^39)*a^3 + O(3^39)*a^2 + O(3^40)*a + (1 + 
        O(3^40)))*$.1 + -(5*3^35 + O(3^38))*a^5 + O(3^38)*a^4 + (10*3^36 + 
        O(3^39))*a^3 + O(3^39)*a^2 - (2701703435345984179 + O(3^40))*a + 
        O(3^40), 1>,
    <(O(3^38)*a^5 + O(3^38)*a^4 + O(3^39)*a^3 + O(3^39)*a^2 + O(3^40)*a + (1 + 
        O(3^40)))*$.1 + (29642867960*3^-1 + O(3^23))*a^5 + O(3^24)*a^4 - 
        (148214339800*3^-1 + O(3^24))*a^3 + O(3^25)*a^2 + (116512378274*3 + 
        O(3^25))*a + O(3^26), 1>,
    <(O(3^38)*a^5 + O(3^38)*a^4 + O(3^39)*a^3 + O(3^39)*a^2 + O(3^40)*a + (1 + 
        O(3^40)))*$.1 + (29642867960*3^-1 + O(3^23))*a^5 + O(3^24)*a^4 - 
        (148214339800*3^-1 + O(3^24))*a^3 + O(3^25)*a^2 + (349537134821 + 
        O(3^25))*a + O(3^26), 1>
]
O(3^38)*a^5 + O(3^38)*a^4 + O(3^39)*a^3 + O(3^39)*a^2 + O(3^40)*a + 1 + O(3^40)