Base \(\Q_{2}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(4\)
Galois group $F_5$ (as 5T3)

Related objects


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Defining polynomial

\(x^{5} + 2\) Copy content Toggle raw display


Base field: $\Q_{2}$
Degree $d$: $5$
Ramification exponent $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{2}(\sqrt{5})$
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: \( x^{5} + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{4} + z^{3} + 1$
Associated inertia:$4$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$F_5$ (as 5T3)
Inertia group:$C_5$ (as 5T1)
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{5} - 2$