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MathGroup Archive 1998

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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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