In the following, I attempt to calculate the weak splitting field of a polynomial over the $$2$$-adics:
from mclf import *
v0 = QQ.valuation(2)
QQ2 = FakepAdicCompletion(QQ,v0)
K.<x> = QQ[]
f = -28*x^6 - 1064*x^4 - 9296*x^2 - 224
L = QQ2.weak_splitting_field(f)
This raises a
ValueError: defining polynomial (x^6 + 38x^4 + 76x^2 + 8) must be irreducible
The same error appears when attempting to calculate
L = WeakPadicGaloisExtension(QQ2,f)
In total, I found seven polynomials f raising a Value Error like this. They are:
$$f_1 = 36 x^6 - 240x^5 + 280x^4 + 832x^3 - 976x^2 - 1984x - 736$$
$$f_2 = -28x^6 - 1064x^4 - 9296x^2 - 224$$
$$f_3 = 28x^6 + 4648x^4 + 2128x^2 + 224$$
$$f_4 = -32x^6 - 304x^4 - 664x^2 - 4$$
$$f_5 = 4x^6 + 152x^4 + 1328x^2 + 32$$
$$f_6 = -32x^6 - 448x^5 - 176x^4 + 64x^3 - 88x^2 + 16x - 4$$
$$f_7 = 32x^6 + 448x^5 + 176x^4 - 64x^3 + 88x^2 - 16x + 4$$
For each of these polynomials, the splitting field over the $$2$$-adics is of degree $$6$$ with Galois group 6T1. Their ramification degrees are all equal to $$2$$, hence their weak splitting fields should be of degree $$2$$.
In the following, I attempt to calculate the weak splitting field of a polynomial over the$$2$$ -adics:
This raises a
The same error appears when attempting to calculate
In total, I found seven polynomials f raising a Value Error like this. They are:
For each of these polynomials, the splitting field over the$$2$$ -adics is of degree $$6$$ with Galois group 6T1. Their ramification degrees are all equal to $$2$$ , hence their weak splitting fields should be of degree $$2$$ .