SNF in non-Bezout domain

hbghlyj

SageMath

1. R.<s> = EquationOrder(x^2 + 5)
3. A = matrix(R,[
4.     [3, 0,0],
5.     [0, 2+2*s, 0],
6.     [0, 0, 4]
7. ])
8. R.ideal([3, 2+2*s]).gens_reduced()

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$s=\sqrt{-5}$
$R=\Bbb Z$
So the ideal $(3,2+2s)$ has reduced generators $(3, s + 1)$, so it is not principal ideal.

1. A.smith_form()

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报错
ArithmeticError:
cannot find lambda, mu such that lambda*d[i,i] + mu*d[j,j] = t

hbghlyj

注意到$(3,4)=(1)$
$A=\pmatrix{3\\&2+2\sqrt{-5}\\&&4}$可以通过初等行、列变换化成$A'=\pmatrix{1\\&2+2\sqrt{-5}\\&&12}$
The matrix $A'$ is in SNF, since $1\mid 2+2\sqrt{-5}$ and $2+2\sqrt{-5}\mid12$[because $12=(2+2\sqrt{-5})(1-\sqrt{-5})$]

1. R.<s> = EquationOrder(x^2 + 5)
2. 12/(2+2*s)

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-s+1

hbghlyj

In non-Bezout domain, not all matrices can be transformed into Smith normal form.
SageMath strictly follows the SNF algorithm of Bezout domain, it fails to find GCD of $3$ and $2+2\sqrt{-5}$, so it fails to find SNF of $A$, however $A$ does have SNF $A'$.

hbghlyj

话说 SageMath 的论坛 ask.sagemath.org 连接超时了, 前天还能打开