They're not prime ideals, but according to the "going up" and "going down" theorems, integral examples of entities have the lying-over and the incomparability property. According to the primitive element theorem, finite separable ones are simple. Adjunction is the process of creating several "sub-" examples of these entities. Those that are both normal and separable are called (*) Galois, and intermediate ones have a bijection with the subgroups of the Galois group according to the first fundamental theorem of Galois theory. A classic example of these entities involves taking the rational numbers and forming a group of numbers of the form a plus b times the square root of C; in this manner the complex numbers can be created from the real numbers. For 10 points, name these objects, which are constructions in which a field is expanded into 
They're not prime ideals, but according to the "going up" and "going down" theorems, integral examples of entities have the lying-over and the incomparability property. According to the primitive element theorem, finite separable ones are simple. Adjunction is the process of creating several "sub-" examples of these entities. Those that are both normal and separable are called (*) Galois, and intermediate ones have a bijection with the subgroups of the Galois group according to the first fundamental theorem of Galois theory. A classic example of these entities involves taking the rational numbers and forming a group of numbers of the form a plus b times the square root of C; in this manner the complex numbers can be created from the real numbers. For 10 points, name these objects, which are constructions in which a field is expanded into a larger field.
They're not prime ideals, but according to the "going up" and "going down" theorems, integral examples of entities have the lying-over and the incomparability property.
field extension [prompt on “fields”]