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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q3 = Qp(3, Prec); x = polygen(QQ) L.<t> = Q3.extension(x^4 + 2*x^3 + 2) K.<a> = L.extension(x^4 + 3*t^2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q3 := pAdicField(3, Prec); K := LocalField(Q3, Polynomial(Q3, [16, 0, 3, 64, 32, 0, 96, 96, 24, 64, 96, 48, 24, 32, 24, 8, 1]));
 

$( x^{4} + 2 x^{3} + 2 )^{4} + 3 x^{2}$ Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content magma:DefiningPolynomial(K);
 

Invariants

Base field: $\Q_{3}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q3;
 
Degree $d$: $16$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$4$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$4$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$12$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{3}$
Root number: $-1$
$\Aut(K/\Q_{3})$ $=$ $\Gal(K/\Q_{3})$: $\OD_{16}$
This field is Galois over $\Q_{3}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[\ ]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:$[2]$
Roots of unity:$240 = (3^{ 4 } - 1) \cdot 3$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, $\Q_{3}(\zeta_{80})$, $\Q_{3}(\zeta_{8}, \sqrt{3})$, $\Q_{3}(\zeta_{8}, \sqrt{\zeta_{8} \cdot 3})$, $\Q_{3}(\zeta_{80}, \sqrt{3})$, $\Q_{3}(\zeta_{8}, \sqrt[4]{\zeta_{8} \cdot 3})$ x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Canonical tower

Unramified subfield:$\Q_{3}(\zeta_{80})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} + 2 x^{3} + 2 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{4} + 3 t^{2} \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^3 + z^2 + 1$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois degree: $16$
Galois group: $\OD_{16}$ (as 16T6)
Inertia group: Intransitive group isomorphic to $C_4$
Wild inertia group: $C_1$
Galois unramified degree: $4$
Galois tame degree: $4$
Galois Artin slopes: $[\ ]$
Galois Swan slopes: $[\ ]$
Galois mean slope: $0.75$
Galois splitting model:not computed