separable extension
A separable extension is one with no “repeated” roots hiding in its building blocks. When you add an element, look at its minimal polynomial; if that polynomial's roots are all distinct — none doubled up — the element is well-behaved. Separability asks this of every element you adjoin.
Precisely, an element algebraic over K is separable if its minimal polynomial has no repeated roots (in an algebraic closure). An extension L/K is separable if every element of L is separable over K. Repeated roots are detected by the derivative: a polynomial has a repeated root exactly when it shares a common factor with its own derivative.
Here is the reassuring part: over the rational numbers, the real numbers, or any field of characteristic zero, every algebraic extension is automatically separable — repeated roots in an irreducible polynomial simply cannot occur. Separability only becomes a genuine concern over certain fields of prime characteristic. Separable plus normal is the precise definition of a Galois extension.
Over the rationals, x^3 - 2 is separable: its three roots — one real cube root of 2 and two complex ones — are all different, and indeed it shares no common factor with its derivative 3·x^2.
x^3 - 2 has three distinct roots, so it is separable.
Over fields of characteristic zero (like Q, R and C) separability comes for free, so beginners can safely ignore it there. It is the subtle second condition — alongside normality — that a Galois extension must satisfy.