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Hello,

Do you really need v to be irreducible in order to do that? Your algorithm looks fine but you can do cleaner using the method quo_rem which returns at once both the quotient and the remainder.

sage: x = polygen(ZZ, 'x')
sage: f = x^6 + 3*x^3 - x^2 + 1
sage: v = x^2 + 1
sage: decomposition = []
sage: q = f
sage: while q:
....:    q,r = q.quo_rem(v)
....:    decomposition.append(r)

And then you can check that you get back the initial polynomial:

sage: sum(a * v^i  for i,a in enumerate(decomposition))
x^6 + 3*x^3 - x^2 + 1

Vincent