Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. It only takes a minute to sign up.
Sign up to join this community
If I input:
Factor[x^2 + x + 1, Extension -> Sqrt[-3]]
Mathematica returns:
1/4 (-I + Sqrt[3] - 2 I x) (I + Sqrt[3] + 2 I x)
The coefficients in the factorization are not in $Q(\sqrt{-3})$. I was expecting something like:
$$ (x - z_3)*(x - (z_3)^2) $$
where $z_3$ is a primitive third root of unity.
Mathematica uses FactorTerms to remove content from factors. Since Sqrt[-3] evaluates to I Sqrt[3], I gets factored out as "content".
FactorTerms[1 + I Sqrt[3] + 2 x]
I (-I + Sqrt[3] - (2 I) x)
You can use AlgebraicNumber to make extension generators atomic.
Factor[x^2 + x + 1, Extension -> AlgebraicNumber[I Sqrt[3], {0, 1}]]
((2 x + AlgebraicNumber[I Sqrt[3], {1, -1}])
(2 x + AlgebraicNumber[I Sqrt[3], {1, 1}])) / 4
ToRadicals[%]//InputForm
((1 - I*Sqrt[3] + 2*x)*(1 + I*Sqrt[3] + 2*x))/4
Factor[x^2 + 2, Extension -> Sqrt[-2]]. Interestingly, the output ofFactor[x^2 + 1, Extension -> Sqrt[-1]]is of the form you were originally expecting. $\endgroup$