Well, one theorem this appeals to is:
$F(a,b) = F(a)(b)$
that is, we can take a field formed from a finite number of adjunctions, and form it "one adjunction at a time".
By definition, the field $F(a,b)$ is the smallest field containing the set $\{a,b\}$ and $F$. Clearly, the field $F(a)(b)$ contains $F$, as it contains $F(a)$, which contains $F$. $F(a)(b)$ also contains $a$, since it contains $F(a)$ and $a \in F(a)$. Finally, $F(a)(b)$ contains $b$, as well, so we clearly have:
$F(a,b) \subseteq F(a)(b)$.
Now $F(a)(b)$ is the smallest field containing $F(a)$, and $b$. But $F(a,b)$ certainly contains $F(a)$, since it contains $F$ and $a$, and $F(a)$ is the smallest field containing $F$ and $a$. Just as clearly, $b \in F(a)(b)$, so we see that:
$F(a)(b) \subseteq F(a,b)$.
Another result this appeals to is:
If $E$ is a finite extension of $F$, and $K$ is a finite extension of $E$, then $K$ is a finite extension of $F$, with:
$[K:F] = [K:E] \ast [E:F]$.
To see this, let $\{u_1,\dots,u_m\}$ be a basis of $E$ over $F$, and let $\{v_1,\dots,v_n\}$ be a basis of $K$ over $E$.
then, for any element $\alpha \in K$, we have:
$\alpha = c_1v_1 + \cdots + c_nv_n$ for some $c_1,\dots,c_n \in E$.
Setting $c_j = b_{1j}u_1 + \cdots + b_{mj}u_m$ for some $b_{ij} \in F$ (for each $j$), which we can do because the $u_i$ form a basis for $E$ over $F$, we get:
$\alpha = b_{11}u_1v_1 + \cdots + b_{m1}u_mv_1 +\cdots + b_{1n}u_1v_n + \cdots + b_{mn}u_mv_n$
which shows that the set $\{b_{ij}: 1 \leq i \leq m, 1 \leq j \leq n\}$ spans $K$ over $F$.
If we set this sum equal to 0, we have (by the linear independence of the $v_j$ over $E$), that in $E$, we have:
$b_{1j}u_1 + \cdots + b_{mj}u_m = 0$ for EACH $j$, and then by the linear independence of the $u_i$ over $F$ we get $b_{ij} = 0$ for each $i$ (and every $j$), which proves linear independence, so we have a basis with $mn$ elements, so:
$[K:F] = mn = nm = [K:E] \ast [E:F]$, since $[K:E] = n$ and $[E:F] = m$.
These results are BASIC (small pun, there) to understanding algebraic extensions of a field (usually we are interested in FINITE algebraic extensions, which are generated by a finite set of elements each of which is algebraic over $F$).
And, why, you might ask, are we interested in such field extensions? So we can solve polynomials, of course! For example, finding a basis of a finite-dimensional vector space over $F$ for which a given $F$-linear transformation is diagonal (which GREATLY simplifies matrix calculations, if it is even possible, which is not always the case), or at least "as diagonal as possible", involves solving a certain polynomial: the characteristic polynomial of that linear transformation. For example, the real matrix:
$\begin{bmatrix}0&-1\\1&0 \end{bmatrix}$
which represents a 90 degree rotation of the plane, has characteristic polynomial $x^2 + 1$, which has no real solutions...so the proper study of such matrices is in the enlarged field of the complex numbers.
It turns out that polynomials shed a great deal of light on the qualitative behavior of linear transformations, but to effectively use this, we need to find the roots of these polynomials, and to do THAT we need to make sure our field is large enough to contain these roots (since irreducible polynomials of degree > 1 exist over some fields).
Moreover, polynomials make great models for more complicated functions, and are very well-behaved. Often we have some "target value" we are concerned about (for a polynomial that measures manufacturing cost this may be the maximum amount we are prepared to spend in said manufacture), which can easily be turned into a question about roots.
As another example, many things in "the real world" are governed by differential equations of order < 3, which in turn can be modeled (approximated) by LINEAR differential equations of order < 3, which in turn involves solving a quadratic to obtain certain parameters. This is the case, for example, with projectile motion (neglecting friction) affected by gravity.
The central theory you are building up to, in your study of field extensions, is a certain correspondence between fields and groups that is rather mind-blowing in and of itself: the Galois correspondence. It turns out that certain "classical" problems, such as solving the general quintic, or squaring the circle, can be shown to be unsolvable using these ideas. It is a bit surprising that groups (as minimally structured as they are) have some insight to offer into the internal structure of fields (which have structure that encompasses most of the math we use up into secondary school...which includes MOST of the math MOST people ever use).
