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All these very explicit theories cannot be extended to work over arbitrary number field.

He is famous for several mathematical discoveries, including his work on number fields.

When K is a number field, the following properties hold.

Major rugby is played on the number 1 field.

Daily Millions had red, white, and blue number fields, in which players chose two numbers of each color.

This is the key to the efficiency of the number field sieve.

Unfortunately, his method cannot be extended to arbitrary number fields.

Given a number field, the class number is often difficult to compute.

An example of a number field that is not monogenic was first given by Dedekind.

However, for more general number fields, the situation becomes more involved, as will be explained below.