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Here is a variation on the answer of rburing. Does it still look correct ?

def subfield(m, L):
    K = CyclotomicField(m, 'z')
    z = K.zeta(m)
    G = K.galois_group()
    power_list = [z**k for k in range(m)]
    convert = ((sigma, power_list.index(sigma(z)))
               for sigma in G)
    G_gens = [g for g, i in convert if i in L]
    return G.subgroup(G_gens).fixed_field()