Factoring Parametric Polynomials

[parisse]

factor does not introduce sqrt of symbols, factorization is done over Q[x,k] here and there x^2-k is irreducible. But you can easily factor x^2-k yourself by hand, or for more complex expressions by running solve(x^2-k=0,x), see that sqrt(k) is required to factor it, then factor(x^2-k,sqrt(k)) will factor the polynomial over Q[sqrt(k)][x]
Note also that canonical_form can help you understand factorization of 2nd order polynomials.