Base \(\Q_{2}\)
Degree \(17\)
e \(17\)
f \(1\)
c \(16\)
Galois group $C_{17}:C_{8}$ (as 17T4)

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Defining polynomial

\(x^{17} + 2\) Copy content Toggle raw display


Base field: $\Q_{2}$
Degree $d$: $17$
Ramification exponent $e$: $17$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $1$
This field is not Galois over $\Q_{2}.$
Visible slopes:None

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: \( x^{17} + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{16} + z^{15} + 1$
Associated inertia:$8$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{17}:C_8$ (as 17T4)
Inertia group:$C_{17}$ (as 17T1)
Wild inertia group:$C_1$
Unramified degree:$8$
Tame degree:$17$
Wild slopes:None
Galois mean slope:$16/17$
Galois splitting model:Not computed