Number Theory Tables, Department of Mathematics and Statistics, UNCG

Number of totally ramified extensions of Q5 of degree 5

Introduction

Polynomial Invariants #Aut Splitting Field Number of
j Ramification Polygon Slopes Residual Polynomials fT eT #Gal Gal Polynomials Extensions
1{(1,1), (5,0)}[ 1/4 ](z+1){1}14{ 20 }{ 5T3 }154·54·5
(z+2){1}14{ 20 }{ 5T3 }15
(z+3){1}14{ 20 }{ 5T3 }15
(z+4){1}14{ 20 }{ 5T3 }15
2{(1,2), (5,0)}[ 1/2 ](z2+1){1}12{ 10 }{ 5T2 }154·54·5
(z2+2){1}22{ 20 }{ 5T3 }15
(z2+3){1}22{ 20 }{ 5T3 }15
(z2+4){1}12{ 10 }{ 5T2 }15
3{(1,3), (5,0)}[ 3/4 ](z+1){1}14{ 20 }{ 5T3 }154·54·5
(z+2){1}14{ 20 }{ 5T3 }15
(z+3){1}14{ 20 }{ 5T3 }15
(z+4){1}14{ 20 }{ 5T3 }15
4{(1,4), (5,0)}[ 1 ](z4+1){1}21{ 10 }{ 5T2 }154·54·5
(z4+2){1}41{ 20 }{ 5T3 }15
(z4+3){1}41{ 20 }{ 5T3 }15
(z4+4){5}11{ 5 }{ 5T1 }55
5{(1,5), (5,0)}[ 5/4 ](z+4){1}14{ 20 }{ 5T3 }5525252