# Polynomials over Quaternions via Determinants Anonymous

I currently trying to get polynomials over quaternions out of determinants of matrices. Example giving below:

Q.<i,j,k> = QuaternionAlgebra(SR, -1, -1)

A=matrix(4,4,[0,j,0,j,j,0,0,j,0,0,0,0,j,j,0,0])

I=matrix.identity(4)

P=(j*A-x*I)

P.det()

But it doesn't work. Any idea?

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What does "determinant" mean if the matrix is defined over a noncommutative ring? In your case you are lucky that P.change_ring(SR).det() makes sense, but that doesn't work in general.

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I do not know much about quaternion algebra, but I do know it is better to avoid involving the symbolic ring:

sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -1, -1)
sage: K.<x> = Q[]
sage: K
Univariate Polynomial Ring in x over Quaternion Algebra (-1, -1) with base ring Rational Field


The variable x now lives in K:

sage: x.parent()
Univariate Polynomial Ring in x over Quaternion Algebra (-1, -1) with base ring Rational Field


and then:

sage: A = matrix(4,4,[0,j,0,j,j,0,0,j,0,0,0,0,j,j,0,0])
sage: I = matrix.identity(4)
sage: P = j*A - x*I
sage: P
[-x -1  0 -1]
[-1 -x  0 -1]
[ 0  0 -x  0]
[-1 -1  0 -x]
sage: P.parent()
Full MatrixSpace of 4 by 4 dense matrices over Univariate Polynomial Ring in x over Quaternion Algebra (-1, -1) with base ring Rational Field
sage: P.det()
x^4 - 3*x^2 + 2*x

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