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Solving polynomial equations over p-adic fields

Hi - as usual I fear my naiv-IT is coming to the fore here, but I have been going around in circles on this for 3 days and I need help please!!

There are 2 basic questions arising from the same thing:

(1) Is "solve" supposed to be implemented for p-adic numbers at all? I can get solutions to things in finite fields but when I try to "lift" them using O(p^n) etc it all goes wrong.

(2) Quite apart from that, why can I not use "solve" using the "variables" (for which I want solutions) as the indeterminates in a polynomial ring over which the equations are already defined? For example, if I define my polynomial ring via:

sage: R.&lt X &gt = Zq(3^4,2);

sage: RAB.&lt a,b &gt = R[];

and if I then try

sage: solve([a+b==6,a-b==2],[a,b])

it tells me that "a is not a valid variable".

 2 No.2 Revision kcrisman 12052 ●37 ●125 ●245

Solving polynomial equations over p-adic fields

Hi - as usual I fear my naiv-IT is coming to the fore here, but I have been going around in circles on this for 3 days and I need help please!!

There are 2 basic questions arising from the same thing:

(1) Is "solve" supposed to be implemented for p-adic numbers at all? I can get solutions to things in finite fields but when I try to "lift" them using O(p^n) etc it all goes wrong.

(2) Quite apart from that, why can I not use "solve" using the "variables" (for which I want solutions) as the indeterminates in a polynomial ring over which the equations are already defined? For example, if I define my polynomial ring via:

sage: R.&lt X &gt R.<X> = Zq(3^4,2);Zq(3^4,2);
sage: RAB.&lt a,b &gt RAB.< a,b> = R[];R[];


and if I then try

sage: solve([a+b==6,a-b==2],[a,b])solve([a+b==6,a-b==2],[a,b])


it tells me that "a is not a valid variable".