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Solution via factorization over $\mathbb{C}$:

def factor_in_t(f):
    D = [0]*(f.degree()+1)
    for r,m in f.roots(ring=CC):
        d = 1 if r==1 else CC(r.argument()/(2*pi)).algdep(1)[1]
        D[d] += m
    ft = {}
    d = f.degree()
    while True:
        while d>0 and D[d]==0:
            d -= 1
        if d==0:
            return ft
        ft[d] = D[d] // euler_phi(d)
        for i in (1..d-1):
            D[d//gcd(i,d)] -= ft[d]

For example, running

R.<t> = ZZ[]
factor_in_t( (t^2 - t + 1)*(t^2 + t + 1)*(t + 1)^2 )

returns a dict {6:1, 2:1}, which is interpreted as $([6]_t)^1\cdot ([2]_t)^1$.

Solution via factorization over $\mathbb{C}$:

def factor_in_t(f):
    D = [0]*(f.degree()+1)
{}
    for r,m in f.roots(ring=CC):
        a = CC(r.argument()/(2*pi)).algdep(1)
        if a[0]==-1:
            D[a[1]] = D.get(a[1],0) + m
    ft = {}
    K = sorted(D.keys())[::-1]
    for k in K:
        if D[k]>0:
            ft[k] = D[k]
            for d = 1 in divisors(k):
                if r==1 else CC(r.argument()/(2*pi)).algdep(1)[1]
d>1:
                    D[d] += m
    ft = {}
    d = f.degree()
    while True:
        while d>0 and D[d]==0:
            d -= 1
        if d==0:
        ft[k]
    return ft
        ft[d] = D[d] // euler_phi(d)
        for i in (1..d-1):
            D[d//gcd(i,d)] -= ft[d]

For example, running

R.<t> = ZZ[]
factor_in_t( (t^2 - t + 1)*(t^2 + t + 1)*(t + 1)^2 )

returns a dict {6:1, 2:1}, which is interpreted as $([6]_t)^1\cdot ([2]_t)^1$.