Properties

Label 29.2.1.1
Base \(\Q_{29}\)
Degree \(2\)
e \(2\)
f \(1\)
c \(1\)
Galois group $C_2$ (as 2T1)

Related objects

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Defining polynomial

\(x^{2} - 29\)  Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $2$
Ramification exponent $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $1$
Discriminant root field: $\Q_{29}(\sqrt{29})$
Root number: $1$
$|\Gal(K/\Q_{ 29 })|$: $2$
This field is Galois and abelian over $\Q_{29}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 29 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{29}$
Relative Eisenstein polynomial:\( x^{2} - 29 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2$ (as 2T1)
Inertia group:$C_2$
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{2} - 29$  Toggle raw display