Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q2 of degree 8

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), (8,0)}[ 1/7 ](z+1){1}37{ 168 }{ 8T36 }1232323
3{(1,3), (2,2), (8,0)}[ 1, 1/3 ](z+1, z2+1){2}23{ 48 }{ 8T23 }2232324
{(1,3), (8,0)}[ 3/7 ](z+1){1}37{ 168 }{ 8T36 }12323
5{(1,5), (2,2), (8,0)}[ 3, 1/3 ](z+1, z2+1){2}23{ 48 }{ 8T24, 8T23 }22242425
{(1,5), (8,0)}[ 5/7 ](z+1){1}37{ 168 }{ 8T36 }22424
7{(1,7), (2,2), (8,0)}[ 5, 1/3 ](z+1, z2+1){2}2323252526
{(1,7), (2,6), (4,4), (8,0)}[ 1 ](z7+z3+z+1){2}41{ 32 }{ 8T19 }22323
{(1,7), (4,4), (8,0)}[ 1 ](z7+z3+1){1}71{ 56 }{ 8T25 }12323
{(1,7), (2,6), (8,0)}[ 1 ](z7+z+1){1}71{ 56 }{ 8T25 }12323
{(1,7), (8,0)}[ 1 ](z7+1){2}31{ 24 }{ 8T13 }22323
9{(1,9), (2,2), (8,0)}[ 7, 1/3 ](z+1, z2+1){2}23{ 48, ...}{ ..., 8T24, 8T23 }24 [.]262627
{(1,9), (2,6), (4,4), (8,0)}[ 3, 1 ](z+1, z6+z2+1){2}31{ 192 }{ 8T38 }222424
{(1,9), (4,4), (8,0)}[ 5/3, 1 ](z+1, z4+1){1}23{ 192 }{ 8T41 }22 [2]2424
{(1,9), (2,6), (8,0)}[ 3, 1 ](z+1, z6+1){2,4}21{ 16, 32 }{ 8T17, 8T8 }23 [3·2]2424
10{(1,10), (2,2), (8,0)}[ 8, 1/3 ](z+1, z2+1){2}2325 [.]272727
11{(1,11), (2,6), (4,4), (8,0)}[ 5, 1 ](z+1, z6+z2+1){2}31{ 24, 192 }{ 8T13, 8T38 }23 [.]252527
{(1,11), (4,4), (8,0)}[ 7/3, 1 ](z+1, z4+1){1,2}23{ 48, 192 }{ 8T24, 8T41 }23 [3·2]2525
{(1,11), (2,6), (8,0)}[ 5, 1 ](z+1, z6+1){2,4}21{ 16, 32 }{ 8T17, 8T9, 8T8, 8T19 }24 [3·22]2525
13{(1,13), (2,6), (4,4), (8,0)}[ 7, 1 ](z+1, z6+z2+1){2}31{ 192 }{ 8T38 }24 [.]262628
{(1,13), (2,10), (4,4), (8,0)}[ 3, 1 ](z3+z+1, z4+1){1}31{ 96 }{ 8T33 }23 [22]2525
{(1,13), (2,6), (8,0)}[ 7, 1 ](z+1, z6+1){2,4}21{ 32 }{ 8T17, 8T18, 8T15, 8T19 }25 [3·23]2626
14{(1,14), (2,6), (4,4), (8,0)}[ 8, 1 ](z+1, z6+z2+1){2}31{ 192 }{ 8T38 }25 [.]272728
{(1,14), (2,6), (8,0)}[ 8, 1 ](z+1, z6+1){2}2126 [25]2727
15{(1,15), (2,10), (4,4), (8,0)}[ 5, 3, 1 ](z+1, z2+1, z4+1){2,4,8}11{ 8, 16, 32 }{ 8T18, 8T9, 8T11, 8T4, 8T19 }26 [25]262628
{(1,15), (2,12), (4,4), (8,0)}[ 3, 4, 1 ](z+1, z2+1, z4+1){1}23{ 192 }{ 8T41 }25 [23]2626
{(1,15), (2,10), (4,8), (8,0)}[ 5, 1, 2 ](z+1, z2+1, z4+1){2}23{ 48, ...}{ ..., 8T23 }26 [24]2626
{(1,15), (4,8), (8,0)}[ 7/3, 2 ](z+1, z4+1){1}23{ 192 }{ 8T41 }24 [23]2626
17{(1,17), (2,10), (4,4), (8,0)}[ 7, 3, 1 ](z+1, z2+1, z4+1){2,4,8}11{ 8, 16, 32 }{ 8T17, 8T6, 8T2, 8T9, 8T5, 8T15, 8T11, 8T8, 8T4 }27 [26]272729
{(1,17), (2,12), (4,4), (8,0)}[ 5, 4, 1 ](z+1, z2+1, z4+1){2}1127 [25]2727
{(1,17), (2,10), (4,8), (8,0)}[ 7, 1, 2 ](z+1, z2+1, z4+1){2}2327 [25]2727
{(1,17), (2,14), (4,8), (8,0)}[ 3, 2 ](z3+z+1, z4+1){1}31{ 96 }{ 8T33 }24 [23]2626
18{(1,18), (2,10), (4,4), (8,0)}[ 8, 3, 1 ](z+1, z2+1, z4+1){2}1128 [26]282829
{(1,18), (2,10), (4,8), (8,0)}[ 8, 1, 2 ](z+1, z2+1, z4+1){2}2328 [26]2828
19{(1,19), (2,12), (4,4), (8,0)}[ 7, 4, 1 ](z+1, z2+1, z4+1){2}11{ 32, 64, ...}{ 8T29, ..., 8T19 }28 [26]282829
{(1,19), (2,14), (4,8), (8,0)}[ 5, 3, 2 ](z+1, z2+1, z4+1){2,4}11{ 16, 32, 64 }{ 8T10, 8T29, 8T18 }27 [3·24]2727
{(1,19), (2,16), (4,8), (8,0)}[ 3, 4, 2 ](z+1, z2+1, z4+1){1}23{ 192 }{ 8T41 }26 [24]2727
20{(1,20), (2,12), (4,4), (8,0)}[ 8, 4, 1 ](z+1, z2+1, z4+1){2,4}11{ 16, 32, ...}{ 8T17, ..., 8T6, 8T15, 8T8 }29 [9·24]292929
21{(1,21), (2,14), (4,8), (8,0)}[ 7, 3, 2 ](z+1, z2+1, z4+1){2}11{ 16, 32, ...}{ ..., 8T6, 8T15, 8T8 }28 [26]282829
{(1,21), (2,16), (4,8), (8,0)}[ 5, 4, 2 ](z+1, z2+1, z4+1){2}11{ 32, 64 }{ 8T29, 8T20, 8T19 }28 [26]2828
22{(1,22), (2,14), (4,8), (8,0)}[ 8, 3, 2 ](z+1, z2+1, z4+1){2}1129 [27]292929
23{(1,23), (2,16), (4,8), (8,0)}[ 7, 4, 2 ](z+1, z2+1, z4+1){2}1129 [27]292929
24{(1,24), (2,16), (4,8), (8,0)}[ 8, 4, 2 ](z+1, z2+1, z4+1){2,4,8}11{ 8, 16, 32, ...}{ 8T7, 8T17, 8T6, ..., 8T15, 8T1, 8T8 }210 [37·23]210210210