Re: Question about irreducible polynomials

quasi wrote:

On Sat, 08 Apr 2006 03:21:28 +0200, Marc Bogaerts
<mbg.DELSPAMnimda@xxxxxxxxx> wrote:

Suppose I have two polynomials f(x) and g((x) with rational coefficients
that have a common irrational root, what can be said about f or g being
irreducible?

If they are both monic (leading coefficient = 1) and not equal, then
they can't both be irreducible.

quasi

The reason I was asking this question was about the following:

Suppose I have a polynomial like f(x)= x^3-c*x²-b*x-a and it's companion
matrix
M=[[0,0,a],
[1,0,b],
[0,1,c]]

When does the algebra generated by M has divisors of zero?

Supposing f has distinct roots in |C then by some transformation T, M
diagolanizes into to D= T*M*T^-1 =

[ [d1,0,0],
[0,d2,0],
[0,0,d3] ]


Now if some element of this algebra turns out to be a divisor of zero this
means one of the d1, d2 or d3 turns out to be a root of some polynomial
g(x), since multiplication of these matrices, multiplying by rationals and
additions only gives diagonal matrices.


Does this mean that M generates a extention field of |Q?


.

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