pp-adic field Q167(167)\Q_{167}(\sqrt{167})

• Label: 167.1.2.1a1.2
• Base: $\Q_{167}$
• Degree: $2$
• e: $2$
• f: $1$
• c: $1$
• Galois group: $C_2$ (as 2T1)

Defining polynomial

$x^{2} + 835$

Invariants

Base field:	$\Q_{167}$
Degree $d$:	$2$
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$1$
Discriminant root field:	$\Q_{167}(\sqrt{167})$
Root number:	$-i$
$\Aut(K/\Q_{167})$ $=$$\Gal(K/\Q_{167})$:	$C_2$
This field is Galois and abelian over $\Q_{167}.$
Visible Artin slopes:	$[\ ]$
Visible Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Jump set:	undefined
Roots of unity:	$166 = (167 - 1)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 167 }$.

Canonical tower

Unramified subfield:	$\Q_{167}$
Relative Eisenstein polynomial:	$x^{2} + 835$

Ramification polygon

Residual polynomials:	$z + 2$
Associated inertia:	$1$
Indices of inseparability:	$[0]$

Invariants of the Galois closure

Galois degree:	$2$
Galois group:	$C_2$ (as 2T1)
Inertia group:	$C_2$ (as 2T1)
Wild inertia group:	$C_1$
Galois unramified degree:	$1$
Galois tame degree:	$2$
Galois Artin slopes:	$[\ ]$
Galois Swan slopes:	$[\ ]$
Galois mean slope:	$0.5$
Galois splitting model:	$x^{2} - 167$