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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q5 = Qp(5, Prec); x = polygen(QQ) L.<t> = Q5.extension(x^4 + 4*x^2 + 4*x + 2) K.<a> = L.extension(x^4 + 5)
 
Copy content magma:Prec := 100; // Default precision of 100 Q5 := pAdicField(5, Prec); K := LocalField(Q5, Polynomial(Q5, [21, 128, 512, 1280, 2208, 2752, 2624, 2048, 1432, 864, 448, 192, 104, 16, 16, 0, 1]));
 

$( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ Copy content Toggle raw display

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

Invariants

Base field: $\Q_{5}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q5;
 
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:$\Q_{5}$
Root number: $-1$
$\Aut(K/\Q_{5})$ $=$ $\Gal(K/\Q_{5})$: $C_4^2$
This field is Galois and abelian over $\Q_{5}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[\ ]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:$[1]$
Roots of unity:$3120 = (5^{ 4 } - 1) \cdot 5$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, $\Q_{5}(\zeta_{624})$, $\Q_{5}(\zeta_{24}, \sqrt{5})$, $\Q_{5}(\zeta_{24}, \sqrt{\zeta_{24} \cdot 5})$, $\Q_{5}(\sqrt[4]{5})$, $\Q_{5}(\sqrt[4]{-5})$, $\Q_{5}(\sqrt[4]{5 \cdot 3})$, $\Q_{5}(\sqrt[4]{5 \cdot 2})$, $\Q_{5}(\zeta_{624}, \sqrt{5})$, $\Q_{5}(\zeta_{24}, \sqrt[4]{5})$, $\Q_{5}(\zeta_{24}, \sqrt[4]{\zeta_{24}^{2} \cdot 5})$

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

Canonical tower

Unramified subfield:$\Q_{5}(\zeta_{624})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{4} + 4 x^{2} + 4 x + 2 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{4} + 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^3 + 4 z^2 + z + 4$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois degree: $16$
Galois group: $C_4^2$ (as 16T4)
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