Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q2 of degree 22

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (2,0), (22,0)}[ 1, 0 ](z+1, z20+1){2}1011211·211·211·2
3{(1,3), (2,0), (22,0)}[ 3, 0 ](z+1, z20+1){2}1011{ 112640, ...}{ 22T34, ... }2211·2211·2211·22
5{(1,5), (2,0), (22,0)}[ 5, 0 ](z+1, z20+1){2}10112311·2311·2311·23
7{(1,7), (2,0), (22,0)}[ 7, 0 ](z+1, z20+1){2}1011{ 112640, ...}{ 22T34, ... }2411·2411·2411·24
9{(1,9), (2,0), (22,0)}[ 9, 0 ](z+1, z20+1){2}10112511·2511·2511·25
11{(1,11), (2,0), (22,0)}[ 11, 0 ](z+1, z20+1){2}1011{ 220, ...}{ 22T6, ... }2611·2611·2611·26
13{(1,13), (2,0), (22,0)}[ 13, 0 ](z+1, z20+1){2}10112711·2711·2711·27
15{(1,15), (2,0), (22,0)}[ 15, 0 ](z+1, z20+1){2}1011{ 112640, ...}{ 22T34, ... }2811·2811·2811·28
17{(1,17), (2,0), (22,0)}[ 17, 0 ](z+1, z20+1){2}10112911·2911·2911·29
19{(1,19), (2,0), (22,0)}[ 19, 0 ](z+1, z20+1){2}1011{ 112640, ...}{ 22T34, ... }21011·21011·21011·210
21{(1,21), (2,0), (22,0)}[ 21, 0 ](z+1, z20+1){2}101121111·21111·21111·211
22{(1,22), (2,0), (22,0)}[ 22, 0 ](z+1, z20+1){2}1011{ 220, ...}{ 22T6, ... }212 [.]11·21211·21211·212