$p$-adic field search results

Degree	Residue characteristic	Galois group	Galois unramified degree
Ramification index	Discriminant exponent	Inertia subgroup	Galois tame degree
Residue field degree	Top slope	Wild inertia subgroup

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Results (8 matches)

 

Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Slope content
7.8.0.1	$8$	x^8 - x + 3	$7$	$1$	$8$	$0$	$C_8$ (as 8T1)	$8$	$1$	$[\ ]^{8}$
7.8.4.1	$8$	x^8 + 14x^6 + 539x^4 + 343x^2 + 60025	$7$	$2$	$4$	$4$	$C_4\times C_2$ (as 8T2)	$4$	$2$	$[\ ]_{2}^{4}$
7.8.4.2	$8$	x^8 + 49x^4 - 1029x^2 + 12005	$7$	$2$	$4$	$4$	$C_8$ (as 8T1)	$4$	$2$	$[\ ]_{2}^{4}$
7.8.6.1	$8$	x^8 + 35x^4 + 441	$7$	$4$	$2$	$6$	$Q_8$ (as 8T5)	$2$	$4$	$[\ ]_{4}^{2}$
7.8.6.2	$8$	x^8 - 49x^4 + 3969	$7$	$4$	$2$	$6$	$D_4$ (as 8T4)	$2$	$4$	$[\ ]_{4}^{2}$
7.8.6.3	$8$	x^8 - 7x^4 + 147	$7$	$4$	$2$	$6$	$C_8:C_2$ (as 8T7)	$4$	$4$	$[\ ]_{4}^{4}$
7.8.7.1	$8$	x^8 + 14	$7$	$8$	$1$	$7$	$D_{8}$ (as 8T6)	$2$	$8$	$[\ ]_{8}^{2}$
7.8.7.2	$8$	x^8 - 7	$7$	$8$	$1$	$7$	$D_{8}$ (as 8T6)	$2$	$8$	$[\ ]_{8}^{2}$