# Unexpected result of PolynomialGCD

Posted 16 days ago
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 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?
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Posted 15 days ago
 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 *) 
 The Automatic setting for Extension will probably suffice for what you want.