Sage 
=================================

### Change the parent back to the polynomial ring
R.<t> = PolynomialRing(QQ) 
f=t^2/t
f.parent()
g=R(f)
g.parent() 

#The following of course gives an error
f=1/t
R(f)


### Using Mignotte's bound 


f = t^5-5*t^4-t + 5
f.norm(2)

### Lifting of a linear factor in the complicated way

R.<t> = PolynomialRing(GF(5))                                                                                                                                                                                                                                           
f=t^3+t+2                                                                                                                                                                                                                                                               
g=f/(t-4)                                                                                                                                                                                                                                                                      
h=t-4                                                                                                                                                                                                                                                                   
g=R(g)
xgcd(g,h)                                                                                                                                                                                                                                                               
output: (1, 4, t + 3)
g=g.change_ring(QuotientRing(ZZ,25*ZZ)) 
h=h.change_ring(QuotientRing(ZZ,25*ZZ))
g=t^2+t+4*t+2


R.<x> = PolynomialRing(QuotientRing(ZZ,25*ZZ))                                                                                                                                                                                                         
g=x^2+4*x+2                                                                                                                                                                                                                                          
h=x-4                                                                                                                                                                                                                                                  
f=x^3+x+2  
a=4
b=(x+3)
t=2*x+3
## Control
g*h-f-5*t

h+5*(-a*t)
s=-2
h2=h+5*(-a*t)+5*s*h
g2=g+5*(-b*t)-5*s*g
g2*h2