pp-adic field Q53(2)\Q_{53}(\sqrt{2})

• Label: 53.2.1.0a1.1
• Base: $\Q_{53}$
• Degree: $2$
• e: $1$
• f: $2$
• c: $0$
• Galois group: $C_2$ (as 2T1)

Defining polynomial

$x^{2} + 49 x + 2$

Invariants

Base field:	$\Q_{53}$
Degree $d$:	$2$
Ramification index $e$:	$1$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$0$
Discriminant root field:	$\Q_{53}(\sqrt{2})$
Root number:	$1$
$\Aut(K/\Q_{53})$ $=$$\Gal(K/\Q_{53})$:	$C_2$
This field is Galois and abelian over $\Q_{53}.$
Visible Artin slopes:	$[\ ]$
Visible Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Jump set:	undefined
Roots of unity:	$2808 = (53^{ 2 } - 1)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 53 }$.

Canonical tower

Unramified subfield:	$\Q_{53}(\sqrt{2})$ $\cong \Q_{53}(t)$ where $t$ is a root of $x^{2} + 49 x + 2$
Relative Eisenstein polynomial:	$x - 53$ $\ \in\Q_{53}(t)[x]$

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois degree:	$2$
Galois group:	$C_2$ (as 2T1)
Inertia group:	trivial
Wild inertia group:	$C_1$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[\ ]$
Galois Swan slopes:	$[\ ]$
Galois mean slope:	$0.0$
Galois splitting model:	$x^{2} - x + 5$