# Properties

 Label 2.5.0.1 Base $$\Q_{2}$$ Degree $$5$$ e $$1$$ f $$5$$ c $$0$$ Galois group $C_5$ (as 5T1)

# Related objects

## Defining polynomial

 $$x^{5} + x^{2} + 1$$ x^5 + x^2 + 1

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $5$ Ramification exponent $e$: $1$ Residue field degree $f$: $5$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{2}$ Root number: $1$ $\card{ \Gal(K/\Q_{ 2 }) }$: $5$ This field is Galois and abelian 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: 2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{5} + x^{2} + 1$$ x^5 + x^2 + 1 Relative Eisenstein polynomial: $$x - 2$$ x - 2 $\ \in\Q_{2}(t)[x]$

## Ramification polygon

The ramification polygon is trivial for unramified extensions.

## Invariants of the Galois closure

 Galois group: $C_5$ (as 5T1) Inertia group: trivial Wild inertia group: $C_1$ Unramified degree: $5$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$
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