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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 + 2*x^3 + 2*x^2 + 2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [2, 0, 2, 2, 0, 0, 0, 0, 1]));
 

\(x^{8} + 2 x^{3} + 2 x^{2} + 2\) Copy content Toggle raw display

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

Invariants

Base field: $\Q_{2}$
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 $c$:$10$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}(\sqrt{5})$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}]$
Visible Swan slopes:$[\frac{1}{3},\frac{1}{3},\frac{1}{2}]$
Means:$\langle\frac{1}{6}, \frac{1}{4}, \frac{3}{8}\rangle$
Rams:$(\frac{1}{3}, \frac{1}{3}, 1)$
Jump set:$[1, 2, 5, 11]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

2.1.4.4a1.1

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: \( x^{8} + 2 x^{3} + 2 x^{2} + 2 \) Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois degree: $48$
Galois group: $\GL(2,3)$ (as 8T23)
Inertia group: $\SL(2,3)$ (as 8T12)
Wild inertia group: $Q_8$
Galois unramified degree: $2$
Galois tame degree: $3$
Galois Artin slopes: $[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},\frac{1}{2}]$
Galois mean slope: $1.3333333333333333$
Galois splitting model:$x^{8} - 6 x^{4} + 4 x^{2} - 3$