There are some inconsistencies in the way `gens`, `gens_dict`, and `gens_dict_recursive` work in rings.
There's some old questions on `gens` in https://groups.google.com/forum/#!msg/sage-devel/BRE2F90oezU/xmKC86SRllEJ;context-place=forum/sage-devel, and some more discussion in trac 22514. The old question and answer suggests that if `R.base_ring()` is not `R`, then `gens` should return the generators for R over its base ring. I hope this means that the smallest subring of `R` containing the base ring and the generators is `R` itself. It is possible that when `R.base_ring()` is `R`, then instead we get generators for `R` in the same sense: the smallest subring of `R` containing the generators is `R` itself. tscrim observed an interesting corner case sage: R = PolynomialRing(QQ, 0, 'x') sage: R.gens() () Here, R is isomorphic, but not identical (according to Sage), to its base ring. We can argue that `()` is indeed the correct answer, but also argue that we should get the same answer as for `QQ.gens()`. For InfinitePolynomialRings, the set of generators is infinite. The current implementation returns a special object as the generator: sage: R = InfinitePolynomialRing(GaussianIntegers(), 'x') sage: R.gens() (x_*,) The returned object is not even an element of the ring, but instead can be indexed to get generators: sage: g = R.gens()[0] sage: g in R False sage: g[4] x_4 sage: g[100] x_100 Admittedly, gens() cannot return an infinite tuple, although it could return something that acted very much like one---which is exactly the value `g` above. It also seems clear than `ngens` should be `+Infinity` rather than 1 as it is now. Method `gens_dict` is apparently meant to return a dictionary that names all the generators. sage: QQ.gens_dict() {'1': 1} sage: GaussianIntegers().gens_dict() {'I': I} sage: ZZ['x', 'y', 'z'].gens_dict() {'x': x, 'y': y, 'z': z} with `R.gens_dict_recursive()` adding the `gens_dict` of `R`, `R.base_ring()`, `R.base_ring().base_ring()`, and so on. sage: QQ['x']['y'].gens_dict() {'y': y} sage: QQ['x']['y'].gens_dict_recursive() {'x': x, 'y': y} sage: GaussianIntegers()['x']['y'].gens_dict() {'y': y} sage: GaussianIntegers()['x']['y'].gens_dict_recursive() {'I': I, 'x': x, 'y': y} The difference between the results for `QQ` and `GaussianIntegers` is odd. In the latter case, the smallest subring of `GaussianIntegers()['x']['y']` containing these three elements is indeed the whole ring, whereas in the former case, `ZZ['x']['y']` contains the returned generators of `QQ['x']['y']`. In part I think this is due to `QQ` being a field; `QQ.gens()` does *not* give a set of generators with respect to just being a ring. The situation for fields is different: sage: K.<cuberoot2> = NumberField(x^3 - 2) sage: L.<cuberoot3> = K.extension(x^3 - 3) sage: S.<sqrt2> = L.extension(x^2 - 2) sage: S Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field sage: S.gens() (sqrt2, cuberoot3, cuberoot2) sage: S.base_field() Number Field in cuberoot3 with defining polynomial x^3 - 3 over its base field sage: S.base_ring() Number Field in cuberoot3 with defining polynomial x^3 - 3 over its base field sage: S.gens_dict() {'cuberoot2': cuberoot2, 'cuberoot3': cuberoot3, 'sqrt2': sqrt2} sage: S.gens_dict_recursive() {'cuberoot2': cuberoot2, 'cuberoot3': cuberoot3, 'sqrt2': sqrt2} Orders behave a bit differently, too: sage: K = NumberField(z^5-z-1, 'a'); K Number Field in a with defining polynomial z^5 - z - 1 sage: O = K.maximal_order(); O Maximal Order in Number Field in a with defining polynomial z^5 - z - 1 sage: O.gens() (1, a, a^2, a^3, a^4) Clearly not a minimal set with respect to being a ring, and indeed `O.ring_generators()` returns just `[a]`. sage: O.gens_dict() {'a': a} Why does the `gens_dict` disagree with `gens`? All of this leaves the intended results of `gens` rather fuzzy. If nothing else, the documentation could be clearer on just how these elements are meant to generate: sage: S.gens? Signature: S.gens() Docstring: Return the generators of this relative number field. EXAMPLES: sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: K.gens() (a, b) sage: Q Univariate Polynomial Ring in x over Eisenstein Integers in Number Field in omega with defining polynomial x^2 + x + 1 sage: Q.gens? Docstring: Return a tuple whose entries are the generators for this object, in order. sage: O.gens? Docstring: Return a tuple whose entries are the generators for this object, in order. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
