On Mon, Feb 6, 2012 at 9:41 AM, Kenneth A. Ribet <[email protected]> wrote:
> Hi All,
>
> This message is about the Sage Reference pages. I'm looking specifically at
> http://www.sagemath.org/doc/reference/sage/rings/finite_rings/constructor.html
> . My comment, which is more of a recommendation, is that it would probably
> be helpful to have a date at the bottom that shows the date of last
> modification. Looking at the material on the Finite Fields page, I can't
> really tell how old it is. (Am I missing something?)
>
> The reason that I was led to this page is because of a discussion in my
> crypto class last week. I mentioned to my class that one knows "by general
> theory" that two finite fields of the same cardinality are isomorphic. On
> the screen at the front of the room, I had two fields F and K of cardinality
> 7^3; they were made with different "moduli" (irreducible polynomials). A
> student asked whether sage could exhibit an isomorphism between the two
> fields. Unfortunately, I didn't know the answer right away. When I read the
> bottom of the page, I was pessimistic that sage could do what the student
> asked:
>
>> While Sage supports basic arithmetic in finite fields some more advanced
>> features for computing with finite fields are still not implemented. For
>> instance, Sage does not calculate embeddings of finite fields yet.
>
The above means that the function "embeddings" just isn't implemented yet:
sage: K.<a> = GF(7^3)
sage: L.<b> = GF(7^3, modulus=(a^2).minimal_polynomial())
sage: L.embeddings(K)
AttributeError
However, one might expect embeddings to be implemented, because it is
implemented for number fields:
sage: x=var('x'); K.<a> = NumberField(x^2+2)
sage: L.<b> = NumberField(x^2+8)
sage: K.embeddings(L)
[
Ring morphism:
From: Number Field in a with defining polynomial x^2 + 2
To: Number Field in b with defining polynomial x^2 + 8
Defn: a |--> 1/2*b,
Ring morphism:
From: Number Field in a with defining polynomial x^2 + 2
To: Number Field in b with defining polynomial x^2 + 8
Defn: a |--> -1/2*b
]
sage:
> It then occurred to me that I could get maps from F to K by factoring over K
> the
> irreducible polynomial corresponding to a generator of F. This works
> beautifully,
> and one gets the required embeddings (which are isomorphisms in this case).
Yes, that's how the function works over number fields under the hood.
It sounds like you should
contribute a patch to add this functionality to Sage (or get one of
your students to). :-).
>
> Ken
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
--
To post to this group, send email to [email protected]
To unsubscribe from this group, send email to
[email protected]
For more options, visit this group at
http://groups.google.com/group/sage-support
URL: http://www.sagemath.org
