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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q2 = Qp(2, Prec); x = polygen(QQ) K.<a> = Q2.extension(x^8 + 8*x^7 + 4*x^6 + 8*x^5 + 2*x^4 + 8*x^3 + 20*x^2 + 16*x + 14)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [14, 16, 20, 8, 2, 8, 4, 8, 1]));
 

\(x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 14\) Copy content Toggle raw display

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

Invariants

Base field: $\Q_{2}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q2;
 
Degree $d$: $8$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$8$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$1$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$25$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}(\sqrt{-2})$
Root number: $i$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[2, 3, \frac{17}{4}]$
Visible Swan slopes:$[1,2,\frac{13}{4}]$
Means:$\langle\frac{1}{2}, \frac{5}{4}, \frac{9}{4}\rangle$
Rams:$(1, 3, 8)$
Jump set:$[1, 2, 4, 16]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.1.4.8b1.6

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

Canonical tower

Unramified subfield:$\Q_{2}$
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 14 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:$1$,$1$,$1$
Indices of inseparability:$[18, 10, 4, 0]$

Invariants of the Galois closure

Galois degree: $64$
Galois group: $C_2\wr C_2^2$ (as 8T31)
Inertia group: $C_2^3:C_4$ (as 8T21)
Wild inertia group: $C_2^3:C_4$
Galois unramified degree: $2$
Galois tame degree: $1$
Galois Artin slopes: $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$
Galois Swan slopes: $[1,2,\frac{5}{2},3,\frac{13}{4}]$
Galois mean slope: $3.8125$
Galois splitting model:$x^{8} - 12 x^{6} + 42 x^{4} - 36 x^{2} - 18$