Factoring an integer inside a ring of integers different from $\mathbb{Z}$

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The ring of integers of the 23rd cyclotomic field is famously not a unique factorisation domain.

Once you have defined the cyclotomic field and its ring of integers,

sage: K = CyclotomicField(23)
sage: L = K.ring_of_integers()

the ring of integers does not have a factor method, as you noticed.

Its elements however do have such a method, but trying to factor L(2) hangs:

sage: L(2).factor()

and you have to interrupt it with Ctrl C.

One thing you can do is factor the ideal generated by 2.

sage: two = L.ideal(2)
sage: two.factor()
(Fractional ideal (2, zeta23^11 + zeta23^9 + zeta23^7 + zeta23^6 + zeta23^5 + zeta23 + 1)) * (Fractional ideal (2, zeta23^11 + zeta23^10 + zeta23^6 + zeta23^5 + zeta23^4 + zeta23^2 + 1))