       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:=
<<AbstractAlgebra`Master`
In:=
IrreduciblePolyOverZpQ[ x^5+x^2+1,2]  (* Verify that the polynomial is
irreducible *)
In:=
F=GF[2^5,IrreduciblePolynomial-> x^5+x^2+1] In:=
G=GenerateGroupoid[{x},Multiplication[F],SizeLimit->31] In:=
subgroups=CyclicSubgroups[G]

Since finite fields are cyclic, the subfields would be the subgroups
that you get from Out that are closed with respect to addition
(other than sums that add to 0).  However, for the example you cite,
(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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