Recover polynomial from 1 root

[Valentin Albillo]

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Hi, Albert Chan:

(09-21-2018 01:17 PM)Albert Chan Wrote:  I were reading Mathematical Universe, by William Dunham

Excellent book, which I bought many years ago (both in Spanish ["El Universo Matemático"] and English. Recommended.

Quote:On page 210, he showed how an algebraic number must be a root of a specific polynomial (with integer coefficient). A hard example, r = sqrt(6) / (5^(1/3) + sqrt(3)), is a solution of ...

4 x^12 - 49248 x^10 - 37260 x^8 - 127440 x^6 + 174960 x^4 - 139968 x^2 + 46656 = 0

How does he do that ? What is the trick to recover polynomial from a single root ?

Any given algebraic number such as your r is the root of an infinite number of polynomials, so the one you quoted is usually the minimal polynomial (which is unique) for that algebraic number.

The minimal polynomial can be found with most advanced software (the function is usually called something like "minpol") which do use lattice reduction algorithms such as LLL, or in more recent times, PSLQ. [Of course it can also be done manually by performing some very tedious but purely algebraic manipulations (raising x-r to suitable powers and isolating radicals at one side, rinse and repeat, until you get rid of all radicals).]

I did include a minpol functionality in version 2.0 of my IDENTIFY program for the HP-71B which does just that, find the minimal polynomial for any given input. If the number is indeed algebraic, the polynomial will have it as one of its roots, else the root will be an approximation to the non-algebraic number given (say Pi).

Quote:BTW, anyone who has HP Prime, can you confirm r really is a root of that polynomial ?

You don't need a Prime for that, an HP-71B will do as well, as will many other advanced HP models.

Regards.
V.
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