# How do I limit calculation to a specific field in general or to reals in particular? [duplicate]

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:= Factor[x^4+1]
Out= 1+x^4


If I add Extention->All that would factor over $\mathbb{C}$, not exactly what I had in mind:

In:= Factor[x^4+1, Extension->All]
Out= (-(-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:= Factor[x^4+1, Extension->Sqrt]
Out=-(-1+Sqrt x-x^2) (1+Sqrt x+x^2)


But I don't want to be guessing the extension or rationals, I want to factor over $\mathbb{R}$:

In:= Factor[x^4+1, Extension->Real]
During evaluation of In:= Factor::nalg: Real is not an explicit algebraic number. >>
Out= 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?