Solving polynomial equations over p-adic fields

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Your second question is fairly easy to answer, though perhaps it's not a fun answer.

sage: type(a)
sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict

sage: solve(a,a)
TypeError: The first argument must be a symbolic expression or a list of symbolic expressions.

solve only works with symbolic expressions, not polynomials, which Sage definitely distinguishes between. (E.g., sin(a) for this a won't work.)

Hi all, I found very curious not to find any example of Hensel's lifting in p-adics on various SAGE pages (maybe because I am too much newbie in maths,Sage,ask,.. ??). I had same experience than others about the "solve()" : discovering than it's not an all purpose solver... Here is my example for Hensel lifting in p-adics. One hidden difficulty was about 'Symbolic Expression' : very difficult to cast one integer p-adic to constant Expression :-)...explaining why code has various "I want to stay inside Integer scope!" parts.

# Sample for 7-adics: search quadratic root of 2

# We define ring of p-adics here...but only for printing end result
# because Hensel lemma algorithm is NOT Taylor tangent approximation algorithm

w = Integer(7) ; q = 5 ; R = Zp(w,q) ; print 'Working in ring ',R

# Define function, symbolic expression over integer
f(x) = x^2 - 2 ; print 'f : ',f

# Compute derivate functions
F0 = f.diff(0) ; F1 = f.diff(1)

# Set one integer local solution of f(x) = 0 
r0 = Integer(3) ; f0r0 = F0(x=r0) ; f1r0 = F1(x=r0)
print 'Is ',r0,' one local solution ? ', (Mod(f0r0,w) == 0) and (Mod(f1r0,w) != 0)

# Get one new local solution from older one at each step of loop
rk = r0 ; wk = Integer(1)
for k in range(q-1):  
    print ; print '********** Step ',k,'***********' ; print

    # Compute k-th power of w
    wk = w * wk

    # Compute value of F0 and F1 for rk
    f0rk = Integer(F0(x=rk)) ; print 'F0(rk) = ', f0rk 
    f1rk = Integer(F1(x=rk)) ; print 'F1(rk) = ', f1rk 

    # Most important part: 
    #  - f0rk / wk is "Integer" division, removing the w^k factor
    #  - negative (-), multiply and inverse_mod operation in ring R
    t = Integer(f0rk / wk) * inverse_mod(-f1rk,w) ; print t
    rk = rk + Integer(t*wk) ; print rk

# Print result
print ; print  'Result in ring R,  f(',R(rk),') = ',R(f(rk))