[GAP Forum] subspace spanned by random subset (of finite field over subfield)

[Max Horn]

Am 11.04.2012 um 09:52 schrieb fvanhove at cage.ugent.be:

> Hello,
> 
> thank you very much for the clear response and the examples.
> 
> I had another question about this: finite fields, seen both as fields and as vector spaces.
> I wanted to take a non-zero element of the bigger field, consider the cyclic subgroup generated by it, and let it act on the cosets of the multiplicative group (so in a sense, I want to consider Singer cycles, which are apparently not yet implemented)
> Curiously, however, there seems to be no command to obtain the multiplicative group of a Galois field?

Try this:

G:=Units(GF(3^6));
H:=Subgroup(G, [Z(3^2)]);

Of course you can use any other unit besides Z(3^2).

It is not quite clear to me which cosets of the multiplicative group you mean (first of, which multiplicative group, do you mean the full unit group? And secondly, cosets in where?) so I can't quite answer the second part of your question.

Cheers,
Max





> 
> Kind regards,
> Frédéric
> 
> 
> Quoting Max Horn <max at quendi.de>:
> 
>> Hello again,
>> 
>> there were (at least) two mistakes in my email:
>> 
>> 1) This
>> 
>>> Dear
>> 
>> should of course have been
>> 
>> "Dear Frédéric"
>> 
>> Sorry! Secondly, as Stefan Kohl pointed out,
>> 
>> [...]
>> 
>>> 
>>> Decomposing w^5 with respect to this basis:
>> 
>> Here, I meant w^6 (and that is what the example code below does).
>> 
>> 
>> Cheers,
>> Max
>> 
>>> 
>>> gap> Coefficients(Basis(W), w^6);
>>> [ Z(3)^0, Z(3)^0, Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ]
>>> 
>>> We verify this:
>>> 
>>> gap> w^0 + w - w^2 + w^4 = w^6;
>>> true
>>> 
>>> 
>>> Hope that helps,
>>> Max
>>> 
>>>> 
>>>> Thanks in advance,
>>>> Kind regards,
>>>> Frédéric
>>>> 
>>>> 
>>>> 
>>>> 
>>>> 
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