factor(x^4+1) and partfrac(1/(x^4+1))

[parisse]

to rombva: I was not clear enough. solve will help you find some algebraic extensions of Q (mainly sqrt()) but it will not help split every polynomial.
x^7+1 can be split with one rootof,
factor(x^7+1,rootof(x^6-x^5+x^4-x^3+x^2-x+1))
but that's not the generic case: in degree n, the extension may be of order n! and that would be too complicated even for small degrees.
You always have the option to do numeric factorization, like this
cfactor(x^7+1.0) (over C) or factor(x^7+1.0) (over R).