# Properties

 Label 37.11.0.1 Base $$\Q_{37}$$ Degree $$11$$ e $$1$$ f $$11$$ c $$0$$ Galois group $C_{11}$ (as 11T1)

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## Defining polynomial

 $$x^{11} - 3 x + 15$$ ## Invariants

 Base field: $\Q_{37}$ Degree $d$: $11$ Ramification exponent $e$: $1$ Residue field degree $f$: $11$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{37}$ Root number: $1$ $|\Gal(K/\Q_{ 37 })|$: $11$ This field is Galois and abelian over $\Q_{37}.$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q_{ 37 }$.

## Unramified/totally ramified tower

 Unramified subfield: 37.11.0.1 $\cong \Q_{37}(t)$ where $t$ is a root of $$x^{11} - 3 x + 15$$ Relative Eisenstein polynomial: $$x - 37$$$\ \in\Q_{37}(t)[x]$ ## Invariants of the Galois closure

 Galois group: $C_{11}$ (as 11T1) Inertia group: trivial Wild inertia group: $C_1$ Unramified degree: $11$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$