What's so special about quadratic extensions?

Question

Reading through chapter 13 "Field Theory" from Dummit and Foote Algebra.

I am wondering why such an emphasis is placed upon "quadratic extensions" of a field F. They state that for any field F (ch(F) not 2) all degree 2 extensions have the form F(squarerootD) read "F adjoin the square root of D" where D is not the square of another element of F.

Does an analogous statement for degree 3 extensions hold? For degree n? Even if those analogous statements do NOT hold for other degrees, am I missing something else about why this statement is important?

Thanks for the help! See below for the text reference:

Answer 1

Here is an example where they used these facts about all degree two extensions. Since all degree two extensions take this form described above, all degree two extensions are Galois (meaning they have full automorphism group). That is already significant, but then they used that fact to show a somewhat surprising result that Galois extensions of a Galois extension need not be Galois. See below: