How to define a homomorphism between two algebras

[Irene]

I have defined two extensions A1 and A2 over a finite field Fp2 with generator b,

A1.<theta>=Fp2.extension(Ep)

A2.<z>=Fp2.extension(Q)

being Ep and Q polynomials.

Now I want to define a homomorphism between those algebras. I have already computed alpha, that is the element in A2 where theta is mapped, but Sage doesn't allow me to define it as:

A1.hom([alpha], A2)

Do you know how to do it?

Irene

[Peter Bruin]

Can you post a complete example?  The following (simple) example works for me (at least in 6.2.beta8):

sage: F=GF(5).extension(2)
sage: A1.<y>=F.extension(x^2+3)
sage: A2.<z>=F.extension(x^2+3)
sage: A1.hom([z],A2)
Ring morphism:
  From: Univariate Quotient Polynomial Ring in y over Finite Field in a of size 5^2 with modulus y^2 + 3
  To:   Univariate Quotient Polynomial Ring in z over Finite Field in a of size 5^2 with modulus z^2 + 3
  Defn: y |--> z

Peter


Op donderdag 24 april 2014 16:55:34 UTC+1 schreef Irene:

[Irene]

Yes, this is the example: 

p=3700001

Fpr=GF(pow(p,2),'b')

b=Fpr.gen()

FFpr.<x>=PolynomialRing(Fpr)

EP= x^6 + (973912*b + 2535329)*x^5 + (416282*b + 3608920)*x^4 + (686636*b + 908282)*x^3 + (2100014*b + 2063451)*x^2 + (2563113*b + 751714)*x + 2687623*b + 1658379

A1.<theta>=Fpr.extension(EP)

Qx=x^6 + (1028017*b + 514009)*x^5 + 2*x^4 + (1028017*b + 514008)*x^3 + 2*x^2 + (1028017*b + 514009)*x + 1

A2.<z>=Fpr.extension(Qx)

alpha=(1636197*b + 1129870)*z^5 + (1120295*b + 3059639)*z^4 + (2637744*b + 3273090)*z^3 + (3564174*b + 890965)*z^2 + (3503957*b + 2631102)*z + 3343290*b + 146187

f=A1.hom([alpha],A2)

[John Cremona]

On 25 April 2014 08:26, Irene <irene....@gmail.com> wrote:
> Yes, this is the example:
>
> p=3700001
> Fpr=GF(pow(p,2),'b')
> b=Fpr.gen()
> FFpr.<x>=PolynomialRing(Fpr)
> EP= x^6 + (973912*b + 2535329)*x^5 + (416282*b + 3608920)*x^4 + (686636*b +
> 908282)*x^3 + (2100014*b + 2063451)*x^2 + (2563113*b + 751714)*x + 2687623*b
> + 1658379
> A1.<theta>=Fpr.extension(EP)
> Qx=x^6 + (1028017*b + 514009)*x^5 + 2*x^4 + (1028017*b + 514008)*x^3 + 2*x^2
> + (1028017*b + 514009)*x + 1
>
> A2.<z>=Fpr.extension(Qx)
> alpha=(1636197*b + 1129870)*z^5 + (1120295*b + 3059639)*z^4 + (2637744*b +
> 3273090)*z^3 + (3564174*b + 890965)*z^2 + (3503957*b + 2631102)*z +
> 3343290*b + 146187
> f=A1.hom([alpha],A2)

This fails because Qx(alpha) is not 0. You need to map theta to a
root of Qx in A2. UNfortunately simple things like

sage: Qx.roots(A1, multiplicities=False)

sage: Qx.change_ring(A1).factor()

fail with a not-implemented error. I think this is because A1 and A2
have not been constructed as fields, though both A1.is_field() and
A2.is_field() return True. It might work to construct GF(p^12)
separately and define isomorphims from both A1 and A2 to it.

Unfortunately you are discovering that the ability of Sage to work
with relative extensions of finite fields is not as good as it should
be. There has been fairly recent work on this, and maybe Peter Bruin
knows what stage that has reached.

John

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[Peter Bruin]

Hello,

In the future (hopefully soon; it is being worked on as I write this, see http://trac.sagemath.org/ticket/14990) Sage will have algebraic closures of finite fields; once we have those, you can use the subfield of the appropriate degree inside the algebraic closure instead of creating fields with the extension() method.

That aside, I agree with John that you can probably avoid some problems by first creating F_{p^12} and then constructing the other fields inside that.

Peter


Op vrijdag 25 april 2014 09:35:05 UTC+1 schreef John Cremona: