The table presents all isomorphism classes of degree 4 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^4 + 2*x^3 + 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 + x^3 + (2 + 2*t)*x^2 + (2 + t)*x + 1 + t^2 | 2 | 2 | 2 | C22 ( 4T2 ) | 4 | 2 | [ ] 2 2 |
x^4 + 2*t | 4 | 1 | 3 | D4 ( 4T3 ) | 8 | 4 | [ ] 4 2 |
x^4 + t | 4 | 1 | 3 | D4 ( 4T3 ) | 8 | 4 | [ ] 4 2 |