Example RngLocA_construct (H49E1)

We can use the 2 step local fields to help us find an irreducible polynomial over a p-adic field which can be used to extend that p-adic field in 1 step.

> P<x> := PolynomialRing(Integers());
> Zp := pAdicRing(7, 50);
> U := UnramifiedExtension(Zp, x^2 + 6*x + 3);
> R := TotallyRamifiedExtension(U, x^3 + 7*x^2 + 7*x + 7);
> L<a> := LocalField(pAdicField(7, 50), MinimalPolynomial(R.1 + U.1, Zp));
> L;
Extension of 7-adic field mod 7^50 by x^6 + (32 + O(7^50))*x^5 + (390 +
    O(7^50))*x^4 + (2284 + O(7^50))*x^3 + (6588 + O(7^50))*x^2 + (8744 +
    O(7^50))*x + 5452 + O(7^50)

We can also use any irreducible polynomial we can find to define a 1 step extension.

> LocalField(pAdicField(7, 50), x^6 - 49*x^2 + 686);
Extension of 7-adic field mod 7^50 by x^6 + O(7^50)*x^5 + O(7^50)*x^4 +
    O(7^50)*x^3 - (7^2 + O(7^52))*x^2 + O(7^50)*x + 2*7^3 + O(7^53)

Example RngLocA_sub (H49E2)

> Qp := pAdicField(5, 20);
> P<x> := PolynomialRing(Qp);
> L<a> := LocalField(Qp, x^4 + 4*x^2 + 2);
> P<x> := PolynomialRing(L);
> LL<aa> := LocalField(L, x^4 + 4*L.1);
> r := (10236563738184*a^3 - 331496727861*a^2 + 10714284669258*a +
> 8590525712453*5)*aa^2
> -12574685904653*a^3 + 19786544763736*a^2 + 4956446023134*a + 37611818678747;
> S, m := sub< LL | r >;
> S;
Extension of Extension of 5-adic field mod 5^20 by x^4 + O(5^20)*x^3 + (4 +
    O(5^20))*x^2 + (4 + O(5^20))*x + 2 + O(5^20) by (O(5^20)*a^3 + O(5^20)*a^2 +
    O(5^20)*a + (1 + O(5^20)))*x^2 + (O(5^20)*a^3 + O(5^20)*a^2 + O(5^20)*a +
    O(5^20))*x + O(5^20)*a^3 + O(5^20)*a^2 + (4 + O(5^20))*a + O(5^20)
> m(S.1);
(O(5^20)*a^3 + O(5^20)*a^2 + O(5^20)*a + O(5^20))*aa^3 + (O(5^20)*a^3 +
    O(5^20)*a^2 + O(5^20)*a + (1 + O(5^20)))*aa^2 + (O(5^20)*a^3 + O(5^20)*a^2 +
    O(5^20)*a + O(5^20))*aa + O(5^20)*a^3 + O(5^20)*a^2 + O(5^20)*a + O(5^20)