Database of Local Fields of positive characteristic

The table presents all isomorphism classes of degree 9 extensions of F3((t)) with discriminant exponent c < 18.

The data includes Defining polynomial, ramification index e, inertia degree f, valuation of discriminant c, Galois group G, Inertia Group I, Galois Slope Content.

 Defining Polynomial   e   f   c   Galois Group   #G   #I   Galois Slope Content 
x^9 + 2*x^3 + 2*x^2 +x + 1 1 9 0 C9 (9T1) 9 1 [ ] 1 9
x^9 + (2 + t + 2*t^2 + t^3)*x^3 + (2*t + t^2 + 2*t^3)*x + 1 + t^3 3 3 9 S3 × C3 (9T4) 18 6 [ 3/2 ] 2 3
x^9 + 2*t^2*x^5 + 2*t^3*x^3 + 2*t^3*x^2 + t^3 3 3 9 (C33 : C3) : C2 (9T22) 162 54 [ 3/2, 3/2, 3/2 ] 2 3
x^9 + 2*t*x^7 + t^2*x^5 + t^2*x^4 + t^3*x^3 + t^3*x^2 + 2*t^3*x + t^3 3 3 9 (C33 : C3) : C2 (9T22) 162 54 [ 3/2, 3/2, 3/2 ] 2 3
x^9 + (2 + 2*t + 2*t^2 + 2*t^3)*x^3 + (t + t^2 + t^3)*x + 1 + t + t^2 + 2*t^3 3 3 9 S3 × C3 (9T4) 18 6 [ 3/2 ] 2 3
x^9 + 2*t^2*x^5 + t^3*x^3 + 2*t^3*x^2 + t^3 3 3 9 (C33 : C3) : C2 (9T22) 162 54 [ 3/2, 3/2, 3/2 ] 2 3
x^9 + 2*t*x^7 + t^2*x^4 + t^3*x^3 + 2*t^3*x + t^3 3 3 9 (C33 : C3) : C2 (9T22) 162 54 [ 3/2, 3/2, 3/2 ] 2 3
x^9 + t*x^7 + t^2*x^5 + 2*t^2*x^4 + 2*t^3*x^3 + t^3*x^2 + t^3*x + t^3 3 3 9 (C33 : C3) : C2 (9T22) 162 54 [ 3/2, 3/2, 3/2 ] 2 3
x^9 + t*x^7 + 2*t^2*x^4 + 2*t^3*x^3 + t^3*x + t^3 3 3 9 (C33 : C3) : C2 (9T22) 162 54 [ 3/2, 3/2, 3/2 ] 2 3
x^9 + 2*t*x^7 + 2*t^2*x^5 + t^2*x^4 + 2*t^3*x^3 + 2*t^3*x^2 + 2*t^3*x + t^3 3 3 9 C32 : S3 (9T13) 54 18 [ 3/2, 3/2 ] 2 3
x^9 + t*x^7 + 2*t^2*x^5 + 2*t^2*x^4 + t^3*x^3 + 2*t^3*x^2 + t^3*x + t^3 3 3 9 C32 : S3 (9T13) 54 18 [ 3/2, 3/2 ] 2 3
x^9 + t*x + t 9 1 9 (C32 : C8) : C2 (9T19) 144 72 [ 9/8, 9/8 ] 8 2
x^9 + 2*t*x + t 9 1 9 (C32 : C8) : C2 (9T19) 144 72 [ 9/8, 9/8 ] 8 2
x^9 + t*x^2 + t 9 1 10 C32 : Q8 (9T14) 72 36 [ 5/4, 5/4 ] 4 2
x^9 + 2*t*x^2 + t 9 1 10 S32 : C2 (9T16) 72 36 [ 5/4, 5/4 ] 4 2
x^9 + t*x^8 + (2*t^2 + 2*t^3)*x^6 + (2*t^2 + 2*t^3)*x^5 + (t^3 + t^4)*x^3 + (t^3 + 2*t^4 + t^5)*x^2 + t^3 + 2*t^4 + t^5 + t^6 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + (t^2 + 2*t^3)*x^6 + t^3*x^5 + t^4*x^4 + (2*t^3 + t^4)*x^3 + (t^4 + 2*t^5)*x^2 + t^3 + t^4 + t^5 + 2*t^6 3 3 12 C9 (9T1) 9 3 [ 2 ] 1 3
x^9 + t*x^8 + 2*t^3*x^6 + 2*t^2*x^5 + t^3*x^2 + t^3 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + 2*t^2*x^7 + 2*t^3*x^6 + t^3*x^5 + 2*t^3*x^4 + 2*t^4*x^3 + t^4*x^2 + t^3 + 2*t^5 + 2*t^6 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + t*x^8 + t^2*x^7 + 2*t^3*x^6 + (2*t^2 + 2*t^3)*x^5 + (t^3 + 2*t^4)*x^4 + 2*t^4*x^3 + (t^3 + 2*t^4 + t^5)*x^2 + t^3 + 2*t^5 + 2*t^6 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + (2*t + t^3)*x^6 + (2*t + t^3)*x^4 + (2 + t + t^2 + 2*t^3)*x^3 + (2*t + t^3)*x^2 + (2*t + 2*t^2 + t^3)*x + 1 + t + t^2 + 2*t^3 3 3 12 S3 × C3 (9T4) 18 3 [ 2 ] 1 6
x^9 + 2*t*x^8 + 2*t^2*x^7 + (2*t^2 + 2*t^3)*x^6 + (t^2 + t^3)*x^5 + (2*t^3 + 2*t^4)*x^4 + (t^3 + t^4)*x^3 + (2*t^3 + t^4)*x^2 + t^3 + 2*t^4 + t^5 + t^6 3 3 12 C9 : C3 (9T6) 27 9 [ 2, 2 ] 1 3
x^9 + 2*t^2*x^7 + 2*t^3*x^6 + 2*t^3*x^4 + t^3 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + 2*t^2*x^7 + 2*t^3*x^6 + (2*t^3 + 2*t^4)*x^4 + t^3 + t^6 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + 2*t*x^8 + t^2*x^7 + t^3*x^6 + t^2*x^5 + t^3*x^4 + 2*t^3*x^2 + t^3 3 3 12 (C33 : C3) : C2 (9T22) 162 27 [ 2, 2, 2 ] 1 6
x^9 + (2*t^2 + 2*t^3)*x^6 + 2*t^3*x^5 + 2*t^4*x^4 + (t^3 + t^4)*x^3 + (2*t^4 + 2*t^5)*x^2 + t^3 + 2*t^4 + t^5 + t^6 3 3 12 C9 (9T1) 9 3 [ 2 ] 1 3
x^9 + t*x^8 + t^2*x^7 + 2*t^3*x^6 + (2*t^2 + 2*t^3)*x^5 + (t^3 + t^4)*x^4 + (t^3 + 2*t^4 + t^5)*x^2 + t^3 + t^6 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + 2*t*x^8 + t^3*x^6 + t^2*x^5 + 2*t^3*x^2 + t^3 3 3 12 (C33 : C3) : C2 (9T22) 162 27 [ 2, 2, 2 ] 1 6
x^9 + t*x^8 + t^2*x^7 + 2*t^3*x^6 + 2*t^2*x^5 + t^3*x^4 + t^3*x^2 + t^3 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + t*x^8 + 2*t^3*x^6 + (2*t^2 + 2*t^3)*x^5 + (t^3 + 2*t^4 + t^5)*x^2 + t^3 + t^6 3 3 12 C3 wr C3 (9T17) 81 27 [ 2, 2, 2 ] 1 3
x^9 + 2*t*x^8 + 2*t^2*x^7 + 2*t^3*x^6 + t^2*x^5 + 2*t^3*x^4 + 2*t^3*x^2 + t^3 3 3 12 C32 : C3 (9T7) 27 9 [ 2, 2 ] 1 3
x^9 + t*x^8 + 2*t^2*x^7 + t^3*x^6 + 2*t^2*x^5 + 2*t^3*x^4 + t^3*x^2 + t^3 3 3 12 C32 : S3 (9T13) 54 9 [ 2, 2 ] 1 6
x^9 + 2*t*x^8 + 2*t^2*x^7 + 2*t^3*x^6 + (t^2 + 2*t^3)*x^5 + (2*t^3 + 2*t^4)*x^4 + 2*t^4*x^3 + (2*t^3 + 2*t^4 + t^5)*x^2 + t^3 + 2*t^5 + 2*t^6 3 3 12 C9 : C3 (9T6) 27 9 [ 2, 2 ] 1 3
x^9 + (t + 2*t^3)*x^6 + (t + 2*t^3)*x^4 + (2 + 2*t + 2*t^2 + t^3)*x^3 + (t + 2*t^3)*x^2 + (t + t^2 + 2*t^3)*x + 1 + 2*t^2 + 2*t^3 3 3 12 C32 (9T2) 9 3 [ 2 ] 1 3
x^9 + 2*t^2*x^7 + t^3*x^6 + 2*t^3*x^4 + t^3 3 3 12 (C33 : C3) : C2 (9T22) 162 27 [ 2, 2, 2 ] 1 6
x^9 + 2*t*x^4 + t 9 1 12 C32 : C4 (9T9) 36 18 [ 3/2, 3/2 ] 2 2
x^9 + 2*t*x^4 + 2*t*x^3 + t 9 1 12 C32 : C6 (9T11) 54 18 [ 3/2, 3/2 ] 2 3
x^9 + 2*t*x^4 + t*x^3 + t 9 1 12 C32 : C6 (9T11) 54 18 [ 3/2, 3/2 ] 2 3
x^9 + t*x^4 + t 9 1 12 S32 (9T8) 36 18 [ 3/2, 3/2 ] 2 2
x^9 + t*x^4 + 2*t*x^3 + t 9 1 12 C32 : C8 (9T15) 72 18 [ 3/2, 3/2 ] 2 4
x^9 + t*x^4 + t*x^3 + t 9 1 12 C32 : C8 (9T15) 72 18 [ 3/2, 3/2 ] 2 4
x^9 + 2*t*x^5 + t 9 1 13 (C32 : C8) : C2 (9T19) 144 72 [ 13/8, 13/8 ] 8 2
x^9 + 2*t*x^6 + t*x^5 + t*x^3 + t 9 1 13 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 5/3 ] 2 3
x^9 + 2*t*x^5 + t*x^3 + t 9 1 13 C32 : D6 (9T18) 108 54 [ 3/2, 3/2, 5/3 ] 2 2
x^9 + t*x^6 + 2*t*x^5 + 2*t*x^3 + t 9 1 13 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 5/3 ] 2 3
x^9 + 2*t*x^5 + 2*t*x^3 + t 9 1 13 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 5/3 ] 2 3
x^9 + 2*t*x^6 + 2*t*x^5 + 2*t*x^3 + t 9 1 13 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 5/3 ] 2 3
x^9 + t*x^5 + t 9 1 13 (C32 : C8) : C2 (9T19) 144 72 [ 13/8, 13/8 ] 8 2
x^9 + t*x^5 + 2*t*x^3 + t 9 1 13 C32 : D6 (9T18) 108 54 [ 3/2, 3/2, 5/3 ] 2 2
x^9 + t*x^5 + t*x^3 + t 9 1 13 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 5/3 ] 2 3
x^9 + t*x^6 + t*x^5 + t*x^3 + t 9 1 13 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 5/3 ] 2 3
x^9 + 2*t*x^8 + t*x^7 + 2*t*x^3 + t + 2*t^2 9 1 15 S3 × C3 (9T4) 18 18 [ 3/2, 2 ] 2 1
x^9 + t*x^8 + t*x^7 + t*x^6 + 2*t*x^3 + t 9 1 15 S3 × C3 (9T4) 18 18 [ 3/2, 2 ] 2 1
x^9 + t*x^8 + t*x^7 + t*x^6 + 2*t*x^3 + t + t^2 9 1 15 S3 × C3 (9T4) 18 18 [ 3/2, 2 ] 2 1
x^9 + 2*t*x^8 + t*x^7 + t*x^3 + t + 2*t^2 9 1 15 S32 (9T8) 36 18 [ 3/2, 2 ] 2 2
x^9 + t*x^7 + t*x^6 + t 9 1 15 (C32 : C8) : C2 (9T19) 144 72 [ 15/8, 15/8 ] 8 2
x^9 + t*x^7 + t 9 1 15 (C32 : C8) : C2 (9T19) 144 72 [ 15/8, 15/8 ] 8 2
x^9 + t*x^7 + 2*t*x^6 + t 9 1 15 (C32 : C8) : C2 (9T19) 144 72 [ 15/8, 15/8 ] 8 2
x^9 + t*x^7 + t*x^3 + t + 2*t^2 9 1 15 C32 : D6 (9T18) 108 54 [ 3/2, 3/2, 2 ] 2 2
x^9 + t*x^8 + t*x^7 + t*x^3 + t 9 1 15 C32 : D6 (9T18) 108 54 [ 3/2, 3/2, 2 ] 2 2
x^9 + t*x^7 + 2*t*x^3 + t + t^2 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + 2*t*x^8 + t*x^7 + 2*t*x^6 + 2*t*x^3 + t 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + t*x^8 + t*x^7 + 2*t*x^3 + t 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + t*x^7 + 2*t*x^3 + t + 2*t^2 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + t*x^7 + 2*t*x^3 + t 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + 2*t*x^8 + t*x^7 + 2*t*x^6 + 2*t*x^3 + t + t^2 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + 2*t*x^8 + 2*t*x^7 + t*x^3 + t 9 1 15 S3 × C3 (9T4) 18 18 [ 3/2, 2 ] 2 1
x^9 + 2*t*x^8 + 2*t*x^7 + t*x^3 + t + 2*t^2 9 1 15 S3 × C3 (9T4) 18 18 [ 3/2, 2 ] 2 1
x^9 + 2*t*x^8 + 2*t*x^7 + t*x^3 + t + t^2 9 1 15 S3 × C3 (9T4) 18 18 [ 3/2, 2 ] 2 1
x^9 + t*x^8 + 2*t*x^7 + 2*t*x^6 + 2*t*x^3 + t + t^2 9 1 15 S32 (9T8) 36 18 [ 3/2, 2 ] 2 2
x^9 + 2*t*x^7 + t 9 1 15 (C32 : C8) : C2 (9T19) 144 72 [ 15/8, 15/8 ] 8 2
x^9 + 2*t*x^7 + 2*t*x^6 + t 9 1 15 (C32 : C8) : C2 (9T19) 144 72 [ 15/8, 15/8 ] 8 2
x^9 + 2*t*x^7 + t*x^6 + t 9 1 15 (C32 : C8) : C2 (9T19) 144 72 [ 15/8, 15/8 ] 8 2
x^9 + 2*t*x^7 + 2*t*x^3 + t 9 1 15 C32 : D6 (9T18) 108 54 [ 3/2, 3/2, 2 ] 2 2
x^9 + 2*t*x^8 + 2*t*x^7 + t*x^6 + 2*t*x^3 + t 9 1 15 C32 : D6 (9T18) 108 54 [ 3/2, 3/2, 2 ] 2 2
x^9 + 2*t*x^7 + t*x^3 + t 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + 2*t*x^7 + t*x^3 + t + 2*t^2 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + 2*t*x^7 + t*x^3 + t + t^2 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + t*x^8 + 2*t*x^7 + t*x^3 + t 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + t*x^8 + 2*t*x^7 + t*x^3 + t + 2*t^2 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + t*x^8 + 2*t*x^7 + t*x^3 + t + t^2 9 1 15 C3 wr S3 (9T20) 162 54 [ 3/2, 3/2, 2 ] 2 3
x^9 + t*x^8 + t*x^6 + t 9 1 16 C32 : C3 (9T7) 27 9 [ 2, 2 ] 1 3
x^9 + t*x^8 + t*x^3 + t + 2*t^2 9 1 16 (C9 : C3) : C2 (9T10) 54 54 [ 3/2, 2, 13/6 ] 2 1
x^9 + t*x^8 + 2*t*x^3 + t + 2*t^2 9 1 16 (C9 : C3) : C2 (9T10) 54 54 [ 3/2, 2, 13/6 ] 2 1
x^9 + 2*t*x^8 + t 9 1 16 S3 × C3 (9T4) 18 9 [ 2, 2 ] 1 2
x^9 + 2*t*x^8 + t + t^2 9 1 16 S3 × C3 (9T4) 18 9 [ 2, 2 ] 1 2
x^9 + 2*t*x^8 + t + 2*t^2 9 1 16 S3 × C3 (9T4) 18 9 [ 2, 2 ] 1 2
x^9 + 2*t*x^8 + 2*t*x^6 + t + 2*t^2 9 1 16 C32 : C8 (9T15) 72 9 [ 2, 2 ] 1 8
x^9 + 2*t*x^8 + t*x^6 + t 9 1 16 C32 : C8 (9T15) 72 9 [ 2, 2 ] 1 8
x^9 + 2*t*x^8 + t*x^3 + t 9 1 16 ((C33 : C3) : C2) : C2 (9T24) 324 162 [ 3/2, 3/2, 2, 13/6 ] 2 2
x^9 + 2*t*x^8 + t*x^6 + 2*t*x^3 + t^2*x^2 + 2*t^2*x + t 9 1 16 ((C33 : C3) : C2) : C2 (9T24) 324 162 [ 3/2, 3/2, 2, 13/6 ] 2 2
x^9 + 2*t*x^8 + 2*t*x^3 + t 9 1 16 ((C33 : C3) : C2) : C2 (9T24) 324 162 [ 3/2, 3/2, 2, 13/6 ] 2 2
x^9 + t*x^8 + t*x^6 + t + 2*t^2 9 1 16 C32 : C3 (9T7) 27 9 [ 2, 2 ] 1 3
x^9 + 2*t*x^8 + t*x^3 + t^2*x + t + 2*t^2 9 1 16 ((C33 : C3) : C2) : C2 (9T24) 324 162 [ 3/2, 3/2, 2, 13/6 ] 2 2
x^9 + 2*t*x^8 + 2*t*x^3 + t + 2*t^2 9 1 16 ((C33 : C3) : C2) : C2 (9T24) 324 162 [ 3/2, 3/2, 2, 13/6 ] 2 2
x^9 + 2*t*x^8 + t*x^3 + t + 2*t^2 9 1 16 ((C33 : C3) : C2) : C2 (9T24) 324 162 [ 3/2, 3/2, 2, 13/6 ] 2 2
x^9 + t*x^8 + t*x^6 + t + t^2 9 1 16 C32 : C3 (9T7) 27 9 [ 2, 2 ] 1 3
x^9 + t*x^8 + t 9 1 16 C32 : C4 (9T9) 36 9 [ 2, 2 ] 1 4
x^9 + t*x^8 + 2*t*x^6 + t 9 1 16 C32 : C6 (9T11) 54 9 [ 2, 2 ] 1 6
x^9 + t*x^8 + t*x^3 + 2*t^2*x + t + 2*t^2 9 1 16 (C9 : C3) : C2 (9T10) 54 54 [ 3/2, 2, 13/6 ] 2 1
x^9 + t*x^8 + t*x^3 + t 9 1 16 (C9 : C3) : C2 (9T10) 54 54 [ 3/2, 2, 13/6 ] 2 1
x^9 + t*x^8 + 2*t*x^3 + t 9 1 16 (C9 : C3) : C2 (9T10) 54 54 [ 3/2, 2, 13/6 ] 2 1
x^9 + t*x^8 + 2*t*x^3 + 2*t^2*x + t 9 1 16 (C9 : C3) : C2 (9T10) 54 54 [ 3/2, 2, 13/6 ] 2 1
inserted by FC2 system