Compute minimal polynomial

[ecir...@gmail.com]

I have two algebraic numbers defined by minimal polynomials:

    x, z = symbols('x, z')

    p = Poly((x-1)*(x-2)*(x-3)*(x-4))

    q = Poly((x-5)*(x-6)*(x-7)*(x-8))

and I would like to compute the sum of these numbers. I [found](http://math.stackexchange.com/a/155132/327863) that I need to "`z=x+y` is a root of the resultant of `P(x)` and `Q(z−x)` (where we take this resultant by regarding `Q` as a polynomial in only `x`)".

I'm totally new to all this algebraic numbers thing so I don't quite understand the advice but I tried:

    resultant(p, q.subs(x, z-x))

but then I got stuck. Please, could someone explain to me:

- I would like to see the steps that lead to the computation of desired minimal polynomial with the help of resultant. I think I defined the two numbers properly but I don't know how to express that `Q(z−x)`.

- As it stands now, the `resultant` function returns bivariate polynomial but I would've assumed that the resulting minimal polynomial should be univariate. How do I get rid of the second variable?

- The above link also says "`P(x) = Q(y) = 0`". Does it mean that `p` and `q` can't be both `Poly((x-1)*(x-2)*(x-3)*(x-4))`? What if I would like to add the same two numbers?

Thank you very much in advance!

[Kalevi Suominen]

On Wednesday, January 4, 2017 at 6:25:02 PM UTC+2, ecir...@gmail.com wrote:

I have two algebraic numbers defined by minimal polynomials:

    x, z = symbols('x, z')

    p = Poly((x-1)*(x-2)*(x-3)*(x-4))

    q = Poly((x-5)*(x-6)*(x-7)*(x-8))

and I would like to compute the sum of these numbers. I [found](http://math.stackexchange.com/a/155132/327863) that I need to "`z=x+y` is a root of the resultant of `P(x)` and `Q(z−x)` (where we take this resultant by regarding `Q` as a polynomial in only `x`)".

I'm totally new to all this algebraic numbers thing so I don't quite understand the advice but I tried:

    resultant(p, q.subs(x, z-x))

The resultant is used to eliminate one variable which has to be given as an argument. So this is what you want:

       resultant(p, q.subs(x, z - x), x)

[ecir...@gmail.com]

On Wednesday, January 4, 2017 at 8:03:34 PM UTC+1, Kalevi Suominen wrote:

The resultant is used to eliminate one variable which has to be given as an argument. So this is what you want:

       resultant(p, q.subs(x, z - x), x) 

Thanks!

 

[Aaron Meurer]

Is there a typo below? Those polynomials aren't minimal polynomials because they aren't irrreducible. 

Aaron Meurer 

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[ecir...@gmail.com]

No, not a typo but an oversight from me.

Is a polynomial "irreducible" if `factor_list(Poly(...))` is exactly of length 1?

[Kalevi Suominen]

On Thursday, January 5, 2017 at 1:08:22 PM UTC+2, ecir...@gmail.com wrote:

No, not a typo but an oversight from me.

Is a polynomial "irreducible" if `factor_list(Poly(...))` is exactly of length 1?

It is irreducible if there is only one factor and its exponent is 1. The square of an irreducible polynomial is not irreducible.

[Aaron Meurer]

You can also check p.is_irreducible. And to be clear, yes, the factors
from factor_list are guaranteed to be irreducible.

Aaron Meurer

> https://groups.google.com/d/msgid/sympy/d47d09c1-4afb-40b3-a13c-9a971057cc18%40googlegroups.com.

[ecir...@gmail.com]

Understood, thank you all!