Properties

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

Related objects

Defining polynomial

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

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 $$x^{2} + x + 1$$ x^2 + x + 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: $C_2$ (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: $x^{2} - x + 1$

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 $x^2+x+1$.