The table presents all isomorphism classes of degree 8 extensions of F3((t)).
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^8 + 2*x^5 + x^4 + 2*x^2 + 2*x + 2 | 1 | 8 | 0 | C8 ( 8T1 ) | 8 | 1 | [ ] 1 8 |
x^8 + t*x^6 + 2*t^4 | 2 | 4 | 4 | C8 ( 8T1 ) | 8 | 2 | [ ] 2 4 |
x^8 + x^7 + (1 + t)*x^6 + x^4 + 2*x^3 + t^3*x^2 + t^3*x + 1 + t^2 + t^3 + t^4 | 2 | 4 | 4 | C4×C2 ( 8T2 ) | 8 | 2 | [ ] 2 4 |
x^8 + t*x^4 + 2*t^2 | 4 | 2 | 6 | C8×C2 ( 8T7 ) | 16 | 4 | [ ] 4 4 |
x^8 + t^2 | 4 | 2 | 6 | Q8 ( 8T5 ) | 8 | 4 | [ ] 4 2 |
x^8 + 2*x^7 + 2*x^6 + 2*x^5 + (1 + 2*t)*x^4 + (1 + 2*t)*x^3 + 2*x^2 + (1 + 2*t)*x + 1 + t + t^2 | 4 | 2 | 6 | D4 ( 8T4 ) | 8 | 4 | [ ] 4 2 |
x^8 + 2*t | 8 | 1 | 7 | QD16 ( 8T8 ) | 16 | 8 | [ ] 8 2 |
x^8 + t | 8 | 1 | 7 | QD16 ( 8T8 ) | 16 | 8 | [ ] 8 2 |