Properties

Label 7.11.10.1
Base \(\Q_{7}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(10\)
Galois group $F_{11}$ (as 11T4)

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Defining polynomial

\(x^{11} + 7\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $11$
Ramification exponent $e$: $11$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{7}(\sqrt{3})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 7 }) }$: $1$
This field is not Galois over $\Q_{7}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial: \( x^{11} + 7 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{10} + 4z^{9} + 6z^{8} + 4z^{7} + z^{6} + z^{3} + 4z^{2} + 6z + 4$
Associated inertia:$10$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$F_{11}$ (as 11T4)
Inertia group:$C_{11}$ (as 11T1)
Wild inertia group:$C_1$
Unramified degree:$10$
Tame degree:$11$
Wild slopes:None
Galois mean slope:$10/11$
Galois splitting model:$x^{11} - 7$