Unexpected result of PolynomialGCD

Questions
1

I recently got an unexpected result to the following:

PolynomialGCD[x^4 + x + 3/(4*2^(2/3)), 4*x^3 + 1]

Output was ⅛, but I expected x + 1/(2^(2/3)) or a multiple thereof, since that polynomial is a factor:

Expand[ (2^(2/3) - 2*2^(1/3) x + 4 x^2) * (x + 1/2^(2/3))]
= 1 + 4x^3

and:

Expand[(3/4 + x/(2*2^(1/3)) - x^2/2^(2/3) + x^3) * (x + 1/2^(2/3))]
= 3/(4 2^(2/3)) + x + x^4

I also don’t quite understand the remark under ‘Properties’ in this description of PolynomialExtendedGCD where it says ‘d is equal to PolynomialGCD[f,g] up to a factor not containing x’. If I try that:

PolynomialExtendedGCD[x^4 + x + 3/(4*2^(2/3)), 4*x^3 + 1, x]
= {2^(1/3) + 2 x, {8/3, -((2 x)/3)}}

Here 2^(1/3) + 2x takes the place of d, and it indeed is a multiple of x + 1/(2^(2/3)). Can someone corroborate that this is a bug, or enlighten me as to why I should expect this behavior? If this is not a bug, how should I interpret the remark in the documentation of PolynomialExtendedGCD I referred to earlier?

2

By default PolynomialGCD does not use any algebraic extension.

In[159]:= Options[PolynomialGCD]

(* Out[159]= {Extension -> None, Modulus -> 0, Trig -> False} *)

It is easy to make it use whatever algebraics appear in the input.

In[160]:= PolynomialGCD[x^4 + x + 3/(4*2^(2/3)), 4*x^3 + 1, 
 Extension -> Automatic]

(* Out[160]= 2^(1/3) + 2 x *)
3

Now that you mentioned this, I was able to find that in the documentation, great!

I guess I still need to get used to how to read the documentation, I was somehow expecting all important options to be visible in the initial blue bit. Thank you for your help! What are the flags I should use if I want ‘d is equal to PolynomialGCD[f,g] up to a factor not containing x’ to hold?

4

The Automatic setting for Extension will probably suffice for what you want.