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, nam
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”]