### That unnameable confusing thing about Field extensions

A first incursion into the concept of field extensions proves to be rather confusing. And I’ll tell you why.

Things which seem difficult, and sometimes impossible to prove, are proved very easily using concepts of vector spaces. For example, we know that $\Bbb{Q}(a_1,a_2,\dots,a_n)$ is a finite extension of $\Bbb{Q}$. Prove that $a_1^2+a_3^2a_5^7$ is also algebraic over $\Bbb{Q}$.

What would generally be expected is that we look up the irreducible polynomials whose zeroes are $a_1,a_3,a_5$, and then from them somehow form a polynomial whose root is $a_1^2+a_3^2a_5^7$. This could prove to be tricky and time-consuming. What would we do if the expression was longer and more complicated?

In comes the concept of vector spaces. By some tricky observations about how field extensions are vector spaces and how the dimension of a vector space always remains constant, we prove that any algebraic (non-radical) combination of the elements $a_1,a_2,\dots,a_n$ will be algebraic over $\Bbb{Q}$.

This seems a little unfair. We’re almost cheating here. Shouldn’t we somehow be proving things through a clever manipulation of polynomials? Yes we are cheating. Because almost every argument boils down to this: the number of basis elements in a finite dimensional vector space remains constant. This is clearly not a theorem about polynomials per se. We’re bringing foreign concepts in, and solving our problems with little insight as to what really is happening.

This IS unfair. With little insight into what really is happening, we are somehow arriving upon a contradiction, and proving terribly powerful things about polynomials. Until we find a more insightul method, we have to make do, I suppose.