

<p>This function tests local conditions. The inputs are: a number field A, a prime p, a uniformizer u for p, and an element a of A. The output is a list of two booleans: the first is true when a is unramified, and the second is true when a is transverse.</p>

{{{id=1|
def local_conditions(A, p, u, a):
    F=A.residue_field(p)
    vp=a.valuation(p)
    b=F(a/u^vp)
    unramified=(vp%3==0)
    transverse=(b.nth_root(3, all=True)!=[])
    return [unramified, transverse]
///
}}}

<p>One random example.</p>

{{{id=2|
A.<i>=NumberField(x^2+1); A
///
Number Field in i with defining polynomial x^2 + 1
}}}

{{{id=11|
p=A.ideal(3*i+8); p
///
Fractional ideal (3*i + 8)
}}}

{{{id=25|
F=A.residue_field(p); F
///
Residue field of Fractional ideal (3*i + 8)
}}}

{{{id=13|
a=A(18); a
///
18
}}}

{{{id=21|
u=A.uniformizer(p); u
///
i + 27
}}}

{{{id=10|
local_conditions(A, p, u, a)
///
[True, False]
}}}

{{{id=26|

///
}}}

{{{id=29|

///
}}}

{{{id=27|

///
}}}