inseparable extension
Most field extensions you meet — anything over the rationals or over a finite field — are well-behaved: every element is the root of a polynomial whose roots are all distinct. An inseparable extension is the pathological opposite. It contains an element whose minimal polynomial has a repeated root, so that element cannot be cleanly distinguished from its 'algebraic twins'. These extensions live only in positive characteristic over imperfect fields, and they are the reason Galois theory needs a separability hypothesis.
Precisely, an algebraic extension L of K is inseparable if some element a in L is inseparable over K, meaning its minimal polynomial is not a separable polynomial — it has a repeated root in an algebraic closure. In characteristic p, the minimal polynomial of such an a always has the form g(x^(p^e)) for some separable g and some e >= 1, and the purely inseparable case is when a^(p^e) lies in K while a does not.
An inseparable extension behaves strangely: there are fewer embeddings into an algebraic closure than the degree would suggest, the separable degree is a proper divisor of the full degree, and the trace form can degenerate. The standard example is F_p(t^(1/p)) over F_p(t), whose only K-embedding into the closure is the identity even though the degree is p.
F_2(t^(1/2)) over F_2(t): the minimal polynomial of t^(1/2) is x^2 - t = (x - t^(1/2))^2, a repeated root, so the extension is purely inseparable of degree 2.
A purely inseparable degree-p extension with only one embedding.
Over a perfect field there are no inseparable algebraic extensions at all; this is essentially the definition of perfect.