UNCG, Department of Mathematics and Statistics, Sebastian Pauli

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

Introduction
Extensions of Q2 of degree 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 32
Extensions of Q2[x]/(x2+x+1) of degree 2, 4, 8 Extensions of Q2[x]/(x4+x+1) of degree 2, 4, 8
Extensions of Q3 of degree 3, 6, 9, 12, 15, 18, 21, 24, 27     Extensions of Q3[x]/(x2+2x+2) of degree 3, 6, 9
Extensions of Q5 of degree 5, 10, 15, 20, 25, 30, 35, 50, Extensions of Q7 of degree 7, 14, 21, 49
Extensions of Q11 of degree 11, 22, 33, Extensions of Q13 of degree 13, 26, 39,
Extensions of Q17 of degree 17, 34 Extensions of Q19 of degree 19,

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