Properties

Label 2.8.12.27
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(12\)
Galois group $C_2^3:(C_7: C_3)$ (as 8T36)

Related objects

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Defining polynomial

\( x^{8} + 4 x^{5} + 4 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $1$
This field is not Galois over $\Q_{2}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{8} + 2 x^{6} + 2 x^{5} - 14 x^{4} - 16 x^{3} - 2 \)

Invariants of the Galois closure

Galois group:$F_8:C_3$ (as 8T36)
Inertia group:$C_2^3:C_7$
Unramified degree:$3$
Tame degree:$7$
Wild slopes:[12/7, 12/7, 12/7]
Galois mean slope:$45/28$
Galois splitting model:$x^{8} - 2 x^{7} + 10 x^{6} - 6 x^{5} + 2 x^{4} + 8 x^{3} - 16 x^{2} + 16 x + 26$