Philosophy of simple field extensions In B. L. van der Waerden's Algebra stuck on the problem 6.9:

The polynomial $f(x) = x^4 + 1$ is irreducible in the field of rationals.
  Adjoin a root $\theta$ and resolve the polynomial in the extended
  field $\mathbb Q (\theta)$ into prime factors.

Seems like I haven't nailed the idea of extending fields, so I'm asking to control my thoughts and help with troubled places.
We can obtain the desired extension by two ways: either we use the fact, which states that we know, in which field would that polynomial have a root (nonsymbolic adjunction), or we can build residue class field modulo that polynomial (symbolic adjunction).
I don't realize what I'm supposed to do in both cases.


*

*In nonsymbolic way, should I just completely factorize $f(x)$ over $\mathbb C$, or am I to find only one (any shall do) root?
I can write
$f(x) = (x^2-i)(x^2+i) = (x-\sqrt i)(x + \sqrt i)(x - \sqrt{-i})(x + \sqrt{-i})$
but what would that give me?

Or I could simply say that $\theta = \sqrt{\sqrt{-1}}$ is an obvious root, divide $f(x)$ by $(x - \theta)$ and think hard what to do next with the quotient $x^3 + \theta x^2 + {\theta}^2 x + {\theta}^3$?

*In symbolic way, I have to find residue class field modulo $f(x)$. Is this possible at all? I haven't seen yet a single example of making that with polynomials.
I'm really sorry for the size of the question. The systematic ignorance reminds of itself. However, I feel, that a proper answer will amend many other problems in my knowledge.
I am really grateful at least for the time you spent on reading it.

I think it's  important to realize what's taken as obvious here and what not.
Obvious: there is a root in some extension, and there is a prime factorization in another extension
Non obvious: all roots are equivalent in the sense that:
Non obvious: the polynomial can be factored in the simple field extension $Q(\theta)$ for any root $\theta$
In particular, the question is probably supposed to be attacked using a more algebraic manner than simply writing down the factorization over $\mathbb{C}$. Knowing the factorization over $\mathbb{C}$ helps figuring out the algebra at play however.
