Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q(δ)≅Q2[x]/(x2+x+1) of degree 4

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (4,0)}[ 1/3 ](z+1){1}13{ 12 }{ 4T4 }1223·223·22
(z+δ){1}13{ 12 }{ 4T4 }122
(z+(δ+1)){1}13{ 12 }{ 4T4 }122
3{(1,3), (2,2), (4,0)}[ 1 ](z3+z+1){1}31{ 12 }{ 4T4 }1229·223·24
(z3+δz+1){1}31{ 12 }{ 4T4 }122
(z3+(δ+1)z+1){1}31{ 12 }{ 4T4 }122
(z3+z+δ){2}21{ 8 }{ 4T3 }222
(z3+δz+δ){2}21{ 8 }{ 4T3 }222
(z3+(δ+1)z+δ){2}21{ 8 }{ 4T3 }222
(z3+z+(δ+1)){2}21{ 8 }{ 4T3 }222
(z3+δz+(δ+1)){2}21{ 8 }{ 4T3 }222
(z3+(δ+1)z+(δ+1)){2}21{ 8 }{ 4T3 }222
{(1,3), (4,0)}[ 1 ](z3+1){4}11{ 4 }{ 4T2 }22223·22
(z3+δ){1}31{ 12 }{ 4T4 }122
(z3+(δ+1)){1}31{ 12 }{ 4T4 }122
5{(1,5), (2,2), (4,0)}[ 3, 1 ](z+1, z2+1){2,4}1124 [3·22]249·243·26
(δz+1, z2+δ){2}1124 [23]24
((δ+1)z+1, z2+(δ+1)){2}1124 [23]24
(z+δ, z2+1){2,4}1124 [3·22]24
(δz+δ, z2+δ){2}1124 [23]24
((δ+1)z+δ, z2+(δ+1)){2,4}1124 [3·22]24
(z+(δ+1), z2+1){2,4}1124 [3·22]24
(δz+(δ+1), z2+δ){2,4}1124 [3·22]24
((δ+1)z+(δ+1), z2+(δ+1)){2}1124 [23]24
6{(1,6), (2,2), (4,0)}[ 4, 1 ](z+1, z2+1){2}1126 [25]263·263·26
(δz+δ, z2+δ){2}1126 [25]26
((δ+1)z+(δ+1), z2+(δ+1)){2}1126 [25]26
7{(1,7), (2,4), (4,0)}[ 3, 2 ](z+1, z2+1){2}1126 [25]263·263·26
(z+δ, z2+1){2}1126 [25]26
(z+(δ+1), z2+1){2}1126 [25]26
8{(1,8), (2,4), (4,0)}[ 4, 2 ](z+1, z2+1){2,4}1128 [9·24]282828