There always exists a subfield of $\mathbb C$ which is a splitting field for $f(x)$ $\in$ $Q[X]$? So I've been studying field theory on my own, and I just started learning about splitting fields. Based on my understandings if a polynomial, $f(x)$ $\in$ $Q[X]$, then there should be always a subfield of $\mathbb C$ which is a splitting field for that polynomial. Is this true? 
Thanks in advance.

Yes. Every polynomial in $\Bbb C[x]$ has a root in $\Bbb C$ (fundamental theorem of algebra), which means that every polynomial has a linear factor; after dividing we get another polynomial in $\Bbb C[x]$ of smaller degree, which also has a root, which means we get another linear factor we can divide by, and we can continue this process until we're left with $1$, and when we collect all of the linear factors we've gathered (possibly with multiplicity) we have a complete factorization (over $\Bbb C$) of the original polynomial. Every polynomial in $\Bbb Q[x]$ is in also a polynomial in $\Bbb C[x]$.
Indeed, every field $k$ has an algebraic closure $K$. Every polynomial in $K[x]$ factors completely over the field $K$. Splitting fields (which algebraic closures are a particular example of) can be explicitly constructed by iteratively adjoining roots of irreducible factors over higher and higher extensions, and this process might need to be done transfinitely using Zorn's lemma; this should be covered in any text on field theory. (Another way I never see discussed anywhere is forming a splitting ring for polynomials using Vieta's formulas and then quotienting by a maximal ideal.)
