Properties

Label 2.2.0.1
Base \(\Q_{2}\)
Degree \(2\)
e \(1\)
f \(2\)
c \(0\)
Galois group $C_2$ (as 2T1)

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Defining polynomial

\(x^{2} + x + 1\) Copy content Toggle raw display

Invariants

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

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_2$ (as 2T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{2} - x + 1$

Additional information

The image of the $2$-adic logarithm on $1$-units is $$ \log_2(1+M) = \{ 2\alpha + 4 \beta t \mid \alpha,\beta\in\Z_2\}$$ where $M$ is the maximal ideal in the ring of integers of this field and $t$ is a root of $x^2+x+1$.