Simple Enlargement Body
Definition 1
If an extension field of satisfies for some , then is called a Simple Extension of .
Explanation
Simply put, can be seen as an expansion by adding just one that was not in . Speaking in terms of the field of real numbers , adding to its extension field results in .
An important fact is that for , if , then all are uniquely represented like this. In this case, is an element of , and thinking about the field of complex numbers as a simple extension of real numbers, it’s easy to see that all complex numbers can be represented for some as
On another note, an interesting example of simple extensions includes Gaussian integers and added to integers, such as Gaussian integers and Eisenstein integers .
Fraleigh. (2003). A first course in abstract algebra(7th Edition): p270. ↩︎