On 23 oct, 21:01, andrew ewart <[email protected]> wrote:
> If i have an element alpha=3^(1/3)+(7^(1/2)*2^(1/4))
> and an ideal I=<a^3-3, b^2-7,c^4-2, alpha-(a+b*c)>
> how do i show the minimum polynomial of alpha lies in the ideal I
> then use it to construct the minumum polynomial of alpha
>
> So far I have:
> P.<a,b,c> = PolynomialRing(QQ)
> alpha=3^(1/3)+(7^(1/2)*2^(1/4))
> I = Ideal(a^3-3,b^2-7,c^4-2,alpha-(a+b*c))
> al=alpha.minpoly()
> but im not sure if QQ should be RR or not and about the implementation
> of the minpoly() command
Hi,
I hope this helps:
sage: P.<a,b,c,al> = QQbar[]
sage: I = Ideal([a^3-3, b^2-7, c^4-2, al-(a+b*c)])
sage: alpha = QQbar(3^(1/3)+(7^(1/2)*2^(1/4)))
sage: am = alpha.minpoly()(al)
sage: am in I
...
True
sage: am
al^12 + (-12)*al^9 + (-294)*al^8 + 54*al^6 + (-14112)*al^5 +
28812*al^4
+ (-108)*al^3 + (-26460)*al^2 + (-345744)*al - 941111
cheers,
Yann
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