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Matrix of a module endomorphism and its char/min poly

  1. Jul 12, 2011 #1

    UD1

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    This is from "Prime factorization of integral Cayley numbers" by H.P. Rehm

    http://archive.numdam.org/ARCHIVE/AFST/AFST_1993_6_2_2/AFST_1993_6_2_2_271_0/AFST_1993_6_2_2_271_0.pdf" [Broken]

    Let A be the algebra of octonions over rationals, C be the maximal order of integral octonions (given for example in Coxeter basis).

    Let M be a Z-module of rank 8, and a in C such that Ma is a submodule of M. (It is obviously of rank 8 over Z, too).

    The goal is to show that [M : Ma] = norm(a)^4, where [M : Ma] is the index of abelian groups.

    This is completely trivial when a is rational (hence integer).

    The question is actually rather generic. It is quite easy to show that if a is in not in Q, the minimal polynomial of the map x -> x*a is f(X)=X^2 - tr(a)X+norm(a). Why is the characteristic polynomial equal to f^4(X)? Why is the determinant of the map equal to plus/minus [M : Ma]? How can one figure out the characteristic polynomial and/or write down a matrix of the map in some basis in terms of a, its trace and norm? Here the bases for M or Ma are not given.
     
    Last edited by a moderator: May 5, 2017
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