For example, suppose I want to factor $x^4+1$ over reals. Calling Factor without options doesn't help because by default it factors over integers:
In[13]:= Factor[x^4+1]
Out[13]= 1+x^4
If I add Extention->All that would factor over $\mathbb{C}$, not exactly what I had in mind:
In[14]:= Factor[x^4+1, Extension->All]
Out[14]= (-(-1)^(1/4)+x) ((-1)^(1/4)+x) (-(-1)^(3/4)+x) ((-1)^(3/4)+x)
If, on the other hand, I manage to guess the right extension of $\mathbb{Q}$ I'll get the answer I want:
In[16]:= Factor[x^4+1, Extension->Sqrt[2]]
Out[16]=-(-1+Sqrt[2] x-x^2) (1+Sqrt[2] x+x^2)
But I don't want to be guessing the extension or rationals, I want to factor over $\mathbb{R}$:
In[17]:= Factor[x^4+1, Extension->Real]
During evaluation of In[17]:= Factor::nalg: Real is not an explicit algebraic number. >>
Out[17]= Factor[1+x^4,Extension->Real]
So how do I specify the field? Even better, can I change something globally in the notebook to select the field I want?
