Fastest way to determine whether a simple extension is Galois, starting from a polynomial

[Ricardo Buring <ricardo.buring@gmail.com>]

Dear pari-users,

I notice that running e.g.

    length(nfgaloisconj(x^29 - 10*x + 13)) == 29

is much faster than

    length(nfgaloisconj(nfinit(x^29 - 10*x + 13))) == 29

Are both of these valid ways to discover that Q[x]/(x^29 - 10*x + 13)
is not Galois? And do these two tests give the same result general?
The documentation of nfgaloisconj only states that passing a
polynomial is allowed; does it also mean that the answer to both
questions is yes?

I am asking because SageMath uses the latter (slower) test for degrees >= 13.

Best regards,
Ricardo