Properties

Label 17.17.17.1
Base \(\Q_{17}\)
Degree \(17\)
e \(17\)
f \(1\)
c \(17\)
Galois group $F_{17}$ (as 17T5)

Related objects

Downloads

Learn more

Defining polynomial

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

Invariants

Base field: $\Q_{17}$
Degree $d$: $17$
Ramification exponent $e$: $17$
Residue field degree $f$: $1$
Discriminant exponent $c$: $17$
Discriminant root field: $\Q_{17}(\sqrt{17})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 17 }) }$: $1$
This field is not Galois over $\Q_{17}.$
Visible slopes:$[17/16]$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 17 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}$
Relative Eisenstein polynomial: \( x^{17} + 17 x + 17 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 16$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$F_{17}$ (as 17T5)
Inertia group:$F_{17}$ (as 17T5)
Wild inertia group:$C_{17}$
Unramified degree:$1$
Tame degree:$16$
Wild slopes:$[17/16]$
Galois mean slope:$287/272$
Galois splitting model:Not computed