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Re: algebra problem, need help fast!!!
*To*: mathgroup at smc.vnet.net
*Subject*: [mg14676] Re: [mg14645] algebra problem, need help fast!!!
*From*: Ken Levasseur <Kenneth_Levasseur at uml.edu>
*Date*: Sun, 8 Nov 1998 21:15:36 -0500
*Organization*: UMass Lowell Mathematical Sciences
*References*: <199811070709.CAA01816@smc.vnet.net.>
*Sender*: owner-wri-mathgroup at wolfram.com
You should be able to get the subfields using AbstractAlgebra`, which is
available at http://www.central.edu/homepages/hibbarda/EAAM/eaam.html.
I am coauthor of these packages with Al Hibbard (Central College). You
might also be able to do the same with the Standard package
FiniteFields.m, but I don't think that any of its functions identifies
subfields. With out packages, there is no single function that
identifies subfields, but you could proceed as follows:
In[1]:=
<<AbstractAlgebra`Master`
In[2]:=
IrreduciblePolyOverZpQ[ x^5+x^2+1,2] (* Verify that the polynomial is
irreducible *)
In[3]:=
F=GF[2^5,IrreduciblePolynomial-> x^5+x^2+1] In[4]:=
G=GenerateGroupoid[{x},Multiplication[F],SizeLimit->31] In[5]:=
subgroups=CyclicSubgroups[G]
Since finite fields are cyclic, the subfields would be the subgroups
that you get from Out[5] that are closed with respect to addition
(other than sums that add to 0). However, for the example you cite,
your answer is obvious from Out[5]. Your question for degree 4 or 6
(for example) yeilds more interesting results.
Ken Levasseur
UMass Lowell
Base_D wrote:
> Does anyone know how to use mathematica to find all the subfields the
> following field:
>
> K = Z/2Z[X]/(f) where is an irreducible polynomial of degree 5.
>
> Any help would be greatly appreciated. Rick
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