field extension
Imagine you only have the rational numbers to work with, and you wish you could write down sqrt(2). It is not there. So you build a bigger number system that contains all the rationals plus sqrt(2) and everything you can make from them by adding, subtracting, multiplying and dividing. That bigger system is a field extension of the rationals: a larger field that holds your original field intact inside it.
Formally, if K is a field and a larger field L contains K (with the same addition and multiplication), we call L a field extension of K, often written L/K (read “L over K” — this is NOT division). K is the base field and L is the extension field. The point is that L gives you room to solve equations or hold numbers that K could not.
Field extensions are the central objects of Galois theory. By measuring how much bigger L is than K — and what symmetries L has that leave K untouched — we turn questions about equations into questions about groups. Almost everything that follows is a refinement of this single idea: enlarge the field, then study the enlargement.
The complex numbers C are a field extension of the real numbers R: every complex number is a + b·i with a, b real, so C contains R and adds the new element i with i^2 = -1.
C/R is a field extension of degree 2.
The slash in L/K is just notation for “L is an extension of K” — never read it as a quotient. The size of an extension is measured by viewing L as a vector space over K (its dimension is the degree).