Hi David,
I can use R instead of V but in this case I can not take the profit of
the instructions that William has been wrote to represent my
polynomial in Normal basis like "span_of_basis()" and "coordinates()".
So what I ultimately want is that if I write these:
sage: k.<a> = GF(2^5, name='a')
sage: R.<x> = k['x']
sage: MyPoly = x*a^8
sage: print MyPoly
(a^3 + a^2 + 1)*x
My results would be in normal basis and the sage just gives me:
a^8*x
instead of :
(a^3 + a^2 + 1)*x
Maybe my example wasn't clear enough so if I change my program to:
sage: k.<a> = GF(2^5, name='a')
sage: V = k.vector_space()
sage: R.<x> = k['x']
sage: PolyRingElm = x*a^8;
sage: print V(PolyRingElm)
I would like to get something like to get something like:
(0,0,0,x,0)
in Normal Basis. It is exactly my problem. I need to get the sage
results in polynomial ring in normal basis. Not just the sage result
in original field in normal basis.
After all I'm addicted to simicolons because of my habit of C
Programming! Is not necessary at all? ;)
Ahmad
On Sep 6, 6:27 am, "David Joyner" <[EMAIL PROTECTED]> wrote:
> On 9/6/07, Ahmad <[EMAIL PROTECTED]> wrote:
>
>
>
>
>
> > Thanks again! I got the idea now. However, there is a problem which
> > sticks me in the first stage and if I can solve it so your code work
> > as prefect for me. The problem is that the instruction
>
> > v(z)
>
> > is not strong enough for my popuse. It works prefect when z is in k =
> > GF(2^5) but it is not working when z is in the polynomial ring of
> > GF(2^5)[x]. I can show it by an example:
>
> > k.<a> = GF(2^5, name='a');
> > V = k.vector_space();
> > R.<x> = k['x'];
>
> > FieldElm= a;
> > PolyRingElm = x*a;
>
> > print V(FieldElm);
> > print V(PolyRingElm);
>
> Use R not V for that:
>
> sage: print R(PolyRingElm)
> a*x
>
> You are coercing into the ring not the vector space.
> Also, why are you using semicolons?
>
>
>
> > And the result is:
>
> > (0, 1, 0, 0, 0)
> > Traceback (most recent call last):
> > File "<stdin>", line 1, in <module>
> > File
> > "/home/amadi/Programmes/Sage/sage-2.8.3-linux-32bit-debian-4.0-i686-
> > Linu\
> > x/sage_notebook/worksheets/admin/0/code/35.py", line 12, in <module>
> > exec compile(ur'print V(PolyRingElm);' + '\n', '', 'single')
> > File
> > "/home/amadi/Programmes/Sage/sage-2.8.3-linux-32bit-debian-4.0-i686-
> > Linu\
> > x/data/extcode/sage/", line 1, in <module>
>
> > File
> > "/home/amadi/Programmes/Sage/sage-2.8.3-linux-32bit-debian-4.0-i686-
> > Linu\
> > x/local/lib/python2.5/site-packages/sage/modules/free_module.py", line
> > 2802, in __call__
> > return FreeModule_generic_field.__call__(self,e)
> > File
> > "/home/amadi/Programmes/Sage/sage-2.8.3-linux-32bit-debian-4.0-i686-
> > Linu\
> > x/local/lib/python2.5/site-packages/sage/modules/free_module.py", line
> > 504, in __call__
> > w = self._element_class(self, x, coerce, copy)
> > File "vector_modn_dense.pyx", line 134, in
> > vector_modn_dense.Vector_modn_dense.__init__
> > TypeError: can't initialize vector from nonzero non-list
>
> > Could you please help me to make the vector space aspect of my finite
> > field works for its polynomial ring as well and give me something
> > like:
>
> > (0, x, 0, 0, 0)
>
> > Thank you in advance!
>
> > Bests,
> > Ahmad
>
> > On Sep 4, 5:53 pm, "William Stein" <[EMAIL PROTECTED]> wrote:
> > > On 9/4/07, David Joyner <[EMAIL PROTECTED]> wrote:
>
> > > > > I have to define two functions below in order to
> > > > > do this. If people think something like this would be generally
> > > > > useful, then it could be made "built in" to SAGE:
>
> > > > I think it would be nice to have in_terms_of_normal_basis
> > > > (of course you need to change "2" to "p" in general).
> > > > However, I don't understand what to_V does that built-in
> > > > coersion doesn't already do:
>
> > > > sage: k.<a> = GF(2^5)
> > > > sage: V = k.vector_space()
> > > > sage: z = (1+a)^17; z
> > > > a^3 + a + 1
> > > > sage: V(z)
> > > > (1, 1, 0, 1, 0)
>
> > > > This seems to be the same output you gave for to_V(z),
> > > > or am I missing something?
>
> > > Hey, good point!
>
> > > Just change to_V(z) to "V(z)" everywhere. Here's a new worksheet:
>
> > > ahmad -- sage-support
> > > system:sage
>
> > > {{{id=0|
> > > k.<a> = GF(2^5)
>
> > > }}}
>
> > > {{{id=1|
> > > k
> > > ///
> > > Finite Field in a of size 2^5
>
> > > }}}
>
> > > {{{id=2|
> > > V = k.vector_space()
>
> > > }}}
>
> > > {{{id=3|
> > > z = (1+a)^17; z
> > > ///
> > > a^3 + a + 1
>
> > > }}}
>
> > > {{{id=6|
> > > B2 = [(a+1)^(2^i) for i in range(k.degree())]
>
> > > }}}
>
> > > {{{id=7|
> > > W = [V(b) for b in B2]
>
> > > }}}
>
> > > {{{id=8|
> > > V.span(W).dimension()
> > > ///
> > > 5
>
> > > }}}
>
> > > {{{id=9|
> > > W0 = V.span_of_basis(W)
>
> > > }}}
>
> > > {{{id=10|
> > > def in_terms_of_normal_basis(z):
> > > return W0.coordinates(z)
>
> > > }}}
>
> > > {{{id=11|
> > > in_terms_of_normal_basis(a+1)
> > > ///
> > > [1, 0, 0, 0, 0]
>
> > > }}}
>
> > > {{{id=12|
> > > in_terms_of_normal_basis(1 + a + a^2 + a^3)
> > > ///
> > > [1, 0, 0, 1, 0]
>
> > > }}}
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