Properties

 Label 17.13.12.1 Base $$\Q_{17}$$ Degree $$13$$ e $$13$$ f $$1$$ c $$12$$ Galois group $C_{13}:C_6$ (as 13T5)

Related objects

Defining polynomial

 $$x^{13} - 17$$

Invariants

 Base field: $\Q_{17}$ Degree $d$ : $13$ Ramification exponent $e$ : $13$ Residue field degree $f$ : $1$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{17}$ Root number: $1$ $|\Aut(K/\Q_{ 17 })|$: $1$ This field is not Galois over $\Q_{17}$.

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: $$x^{13} - 17$$

Invariants of the Galois closure

 Galois group: $D_{13}:C_3$ (as 13T5) Inertia group: $C_{13}$ Unramified degree: $6$ Tame degree: $13$ Wild slopes: None Galois mean slope: $12/13$ Galois splitting model: Not computed