[PariGP] arithmetic modulo polynomials - best syntax question

["Dr. Wolfgang Lindner" <lindnerw@t-online.de>]

Dear group,

testing congruence modulo a polynomial I try to show that:

        in Z/5Z[X] the polynomials f=1+x+x^2  and  g=1-x+3*x^3
        are _not_ congruent modulo p=X^2.

I like the very straightforward and compact syntax of Pari/GP.
So I tried to show the above task on several ways.
(Sorry to ask such rudimental questions but I am retired and there is
no one else that I could ask.)

--------------------------- Here is my CODE:

f=1+x+x^2
g=1-x+3*x^3

f5=f*Mod(1,5)
g5=g*Mod(1,5)

Mod(f5,x^2)
Mod(g5,x^2)

Mod(f,x^2)
Mod(g,x^2)

\\  1. Test
Mod(Mod(f,x^2),5)
Mod(Mod(g,x^2),5)

\\ 2. Test
Mod(f,x^2)*Mod(1,5)
Mod(g,x^2)*Mod(1,5)

\\ 3. per definition
(f-g) % x^2


------------------------- HERE is the ANSWER of PariGP:

x^2 + x + 1
3*x^3 - x + 1

Mod(1, 5)*x^2 + Mod(1, 5)*x + Mod(1, 5)
Mod(3, 5)*x^3 + Mod(4, 5)*x + Mod(1, 5)

Mod(Mod(1, 5)*x + Mod(1, 5), x^2)
Mod(Mod(4, 5)*x + Mod(1, 5), x^2)

Mod(x + 1, x^2)
Mod(-x + 1, x^2)

Mod(Mod(1, 5)*x + Mod(1, 5), x^2)
Mod(Mod(4, 5)*x + Mod(1, 5), x^2)

Mod(Mod(1, 5)*x + Mod(1, 5), x^2)
Mod(Mod(4, 5)*x + Mod(1, 5), x^2)

Q: - is the test syntax ok?
    - or are there better suited solutions/formulations in Pari/GP ?


Best
Wolfgang
- - -
Dr. Wolfgang Lindner
Leichlingen, Germany