The table presents all isomorphism classes of degree 4 extensions of F5((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^4 + 4*x^2 + 4*x + 2 | 1 | 4 | 0 | C4 (4T1) | 4 | 1 | [ ] 1 4 |
x^4 + t*x^2 + 2*t^2 | 2 | 2 | 2 | C4 (4T1) | 4 | 2 | [ ] 2 2 |
x^4 + 3*x^3 + 2*t*x^2 + (1 + 3*t)*x + 4 + 2*t + t^2 | 2 | 2 | 2 | C22 (4T2) | 4 | 2 | [ ] 2 2 |
x^4 + 3*t | 4 | 1 | 3 | C4 (4T1) | 4 | 4 | [ ] 4 1 |
x^4 + 4*t | 4 | 1 | 3 | C4 (4T1) | 4 | 4 | [ ] 4 1 |
x^4 + 2*t | 4 | 1 | 3 | C4 (4T1) | 4 | 4 | [ ] 4 1 |
x^4 + t | 4 | 1 | 3 | C4 (4T1) | 4 | 4 | [ ] 4 1 |