normal extension
A normal extension is one with no “half-captured” roots. The rule is all-or-nothing: if an irreducible polynomial over the base field manages to sneak even one root into the extension, then the extension is forced to contain all of that polynomial's roots. No root is left stranded outside.
Precisely, an algebraic extension L/K is normal if every irreducible polynomial in K[x] that has at least one root in L splits completely into linear factors over L. Equivalently, L is the splitting field over K of some family of polynomials. So normality is exactly the “closed under sibling roots” condition.
Normality is one of the two ingredients of a Galois extension; the other is separability. Without normality, the Galois group can be too small to see the full picture, because some symmetries that ought to permute roots have nowhere to send them. Adjoining a single cube root of 2 to the rationals is the classic non-normal example: the other two (complex) cube roots are missing.
Q(sqrt(2))/Q is normal: the only irreducible polynomial brought in is x^2 - 2, whose other root -sqrt(2) is already present. But Q(cbrt(2))/Q is not normal: x^3 - 2 has a root there, yet its two complex roots are missing.
Q(sqrt(2)) is normal; Q(cbrt(2)) is not.