[GAP Forum] Looking for proofs of statements that always seem true in GAP calculations
mbg nimda
mbg.nimda at gmail.com
Wed Oct 27 12:34:09 BST 2010
Dear Forum members,
GAP allows me to make the following explicit construction of a commutative
matrix algebra, starting with a separable and irreducible polynomial p(x)
of degree n over the rationals.
I call this algebra the "companion algebra" because it is a generalization
of the companion matrix (and the algebra it generates).
The first step is the construction of the algebra A1 generated by the
companion
matrix M of the polynomial p. This is a commutative algebra of n x n
matrices,
containing a "root" of the polynomial p, namely M.
The second step is to consider the polynomial q(x) = (p(x) - M) / (x - M),
viewed as a polynomial with coefficients in A1. Let N be the companion
matrix
of q. This is a (n-1) x (n-1) block matrix with n x n matrices as entries,
yielding n(n-1) x n(n-1 matrices over the rationals. The matrix M is
"promoted"
to a "scalar" (n-1) x (n-1) block matrix and the algebra A2 is constructed
as
the algebra generated by M and N.
The following steps are simply a repetition of the construction done in the
second step giving rise to algebras A2, ... and "roots" P,Q,...
The construction ends when the remaining polynomial is of the second degree.
The final algebra An is the "companion algebra" announced above.
As an attachment to this post I include the function "companionalgebra"
that explicitely constructs this algebra, providing a basis for this
algebra as well as two generators "phi" and "psi" of the symmetric group Sn,
as automorphisms of An expressed as matrices in this basis.
I can prove some of the following statements but not all of them, but they
seem to be valid in the few dozens of polynomials I tried.
a) An is isomorphic to a direct product of n!/k copies of the splitting
field F of the polynomial, where k is the order of the Galois group of the
polynomial p.
b) There is a subgroup G of the group of automorphisms of An that is
isomorphic
to the symmetric group Sn.
c) There is a basis in An such that the group mentioned in b) acts
transitively
on this basis.
d) The action of G on this basis is analogous to the action of G on the
right cosets of a subgroup H of G which is isomorphic to the Galois group
of p.
I don't know if the construction of the companion algebra is of any
practical
use, but I think it is valid for educational purposes. That is is not
completely
useless can be seen in a posting I made in the group sci.math where I
express
in radicals a root of a quintinc that has the dihedral group D(10) as a
Galois
group:
http://groups.google.com/group/sci.math/browse_thread/thread/9f234150358ce3e7?pli=1
So I home somebody has any hint in the proofs of above statements.
Marc Bogaerts.
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