Properties

Label 17.1.5.4a1.1
Base Q17\Q_{17}
Degree 55
e 55
f 11
c 44
Galois group F5F_5 (as 5T3)

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Defining polynomial

x5+17x^{5} + 17 Copy content Toggle raw display

Invariants

Base field: Q17\Q_{17}
Degree dd: 55
Ramification index ee: 55
Residue field degree ff: 11
Discriminant exponent cc: 44
Discriminant root field: Q17(3)\Q_{17}(\sqrt{3})
Root number: 11
Aut(K/Q17)\Aut(K/\Q_{17}): C1C_1
This field is not Galois over Q17.\Q_{17}.
Visible Artin slopes:[ ][\ ]
Visible Swan slopes:[][]
Means: \langle\ \rangle
Rams:( )(\ )
Jump set:undefined
Roots of unity:16=(171)16 = (17 - 1)

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and Q17\Q_{ 17 }.

Canonical tower

Unramified subfield:Q17\Q_{17}
Relative Eisenstein polynomial: x5+17 x^{5} + 17 Copy content Toggle raw display

Ramification polygon

Residual polynomials:z4+5z3+10z2+10z+5z^{4} + 5 z^{3} + 10 z^{2} + 10 z + 5
Associated inertia:44
Indices of inseparability:[0][0]

Invariants of the Galois closure

Galois degree: 2020
Galois group: F5F_5 (as 5T3)
Inertia group: C5C_5 (as 5T1)
Wild inertia group: C1C_1
Galois unramified degree: 44
Galois tame degree: 55
Galois Artin slopes: [ ][\ ]
Galois Swan slopes: [][]
Galois mean slope: 0.80.8
Galois splitting model:x517x^{5} - 17