Polynomials over local fields can be factored and their roots computed.
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.
For a polynomial f over a general local field, return a precision at which the factorization of f as given by Factorization(f) will be Hensel liftable to the correct factorization.The precision returned is not guaranteed to be enough to obtain a factorization of the polynomial. It may be that a correct factorization cannot be found at that precision but may be possible with a little more precision.
The roots of the polynomial f over the general local field R where R is taken to be the coefficient ring of f if it is not given.
> 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)