# Question:Wrong minimal polynomials?

## Question:Wrong minimal polynomials?

Maple 2024

Consider the following exact algebraic number

 > restart;
 > kernelopts('version'):
 > Physics:-Version():
 >
 >
 memory used=0.97MiB, alloc change=12.00KiB, cpu time=31.00ms, real time=27.00ms, gc time=0ns
 (1)
 >
 (2)
 >
 memory used=70.25GiB, alloc change=48.00MiB, cpu time=23.63m, real time=21.81m, gc time=3.18m
 (3)
 >
 (4)

I would like to find its minimal polynomial (without a priori knowledge).

According to the documentation,

if expr is an exact algebraic number, and n and acc are not given, then `PolynomialTools:-MinimalPolynomial(expr)` will call `evala/Minpoly` to compute an exact minimal polynomial of expr. If a name is not specified for the variable x, then _X will be used.

Regretfully, it is easy to see that the minimal polynomial of  cannot be the returned . And when I invoke evala@Minpoly directly, the result is still not correct (and this evaluation takes a rather long time).
Another help page mentions that:

the call `mp := evala(Minpoly(expr, _X))` computes the monic minimal polynomial of a in the variable _X over the field of rational numbers (or multivariate rational functions); the resulting polynomial will not contain any algebraic numbers or functions.

However, as `type(mp, polynom(rational, _X))` gives false, we know that mp cannot be the desired minimal polynomial of either.
So, what is the proper way to compute the minimal polynomial of  in Maple?

Code:

```use alpha=1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345,index=2)-1/2)**2) in
expr:=(1+alpha)*sqrt(1-alpha**2)+(3+4*alpha)/12*sqrt(3-4*alpha**2)+2*(1+alpha)/3*sqrt(2*(1+alpha)*(1-2*alpha))+(1+2*alpha)/6*sqrt(2*((1-alpha)**2-3*alpha**2))
end:
CodeTools:-Usage(PolynomialTools:-MinimalPolynomial(expr));```

Of note, the minimal polynomial of an algebraic number  is the unique irreducible monic polynomial of smallest degree  with rational coefficients such that  and whose leading coefficient is 1

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