A cyclic extension is a field extension where the Galois group is cyclic. This means that there is an element in the Galois group that generates the entire group, similar to how a clock's hand moves around a circle.
One example of a cyclic extension is the extension of the rational numbers by the nth root of unity, where n is a positive integer. This extension is denoted by Q(ζn), where ζn is a primitive nth root of unity.
To prove that an extension is cyclic, you need to show that the Galois group of the extension is generated by a single element. This can be done by finding an element in the Galois group that generates the entire group, or by showing that the Galois group is isomorphic to a cyclic group.
Proving that an extension is cyclic can give insight into the structure of the extension and its Galois group. It can also help in the study of other fields and their properties, as cyclic extensions have many interesting and useful properties.
The cyclic extension of finite Galois group L/F is one of the possible types of extensions that can be obtained through the Galois correspondence. It is a special case where the Galois group is cyclic, and it is useful in understanding the relationship between the field extensions and their Galois groups.