Totally Ramified p-adic Fields of Degree 2p
The files below contain invariant data for isomorphism classes of totally ramified extensions of the p-adic numbers of degree 2p for p>7. The data is in the format: [poly, j, aut, e, f, res, r, tame, h, gal], where
- poly is an Eisenstein polynomial defining the extension
- the p-adic valuation of the discriminant is 2p+j-1
- aut is the number of automorphisms of the extension
- e is the ramification index of the fixed field of the Sylow p-subgroup of poly's Galois group
- f is the inertia degree of the fixed field of the Sylow p-subgroup of poly's Galois group
- res is the residual polynomial, defined modulo p, associated with the first segment of the ramification polygon of poly
- r is an integer such that the fixed field of the Sylow p-subgroup of poly's Galois group is defined by x^{e}-z^{r}p over the unramified extension of the p-adic numbers of degree f
- tame is the Galois group of the fixed field of the Sylow p-subgroup of poly's Galois group, identified in the Small Groups Library in GAP/Magma
- the extension has a unique degree p subfield if and only if poly is an even polynomial. Let g(x) be the polynomial obtained by halving the exponents poly. The Galois group of g(x) has order ph. Otherwise, h=0
- gal is the Galois group of poly, identified in the Transitive Groups Library in GAP/Magma
Data
Wildly Ramified p-adic Fields
The files below contain invariant data for isomorphism classes of degree n extensions of the p-adic numbers where 11<n<16. The data is in the format: [e, d, f, g], where
- e is the ramification index of the extension
- d is the p-adic valuation of the discriminant
- f is a polynomial defining the extension
- g is the Galois group of f, identified in the Transitive Groups Library in GAP/Magma
Data
- Degree 12
- Degree 14
- Degree 14