R.<t> = Integers(49)[]
f=t^2+t+5
f

myroots=f.roots(multiplicities=False)
myroots

# new poly

R.<x> = ZZ[]
f=x^3+2
f

var('p')
pp = [p^j for j in range(1,6)]
pp

qq = [f.change_ring(Integers(5^j)).roots(multiplicities=false)[0] for j in range(1,6)]
qq

# new poly

p=3

ff = x^4 -7*x^3+2*x^2+2*x+1
ff

ff(2)

ff(2) % 3

diff(ff,x)

diff(ff,x)(2) % 3

ff.change_ring(Integers(p^1)).roots(multiplicities=false)

ff.change_ring(Integers(p^2)).roots(multiplicities=false)

ff.change_ring(Integers(p^3)).roots(multiplicities=false)