Definitions
The following are equivalent definitions of an algebraic integer. Let K be a number field (i.e., a finite extension of ), in other words, for some by the primitive element theorem.
- is an algebraic integer if there exists a monic polynomial such that .
- is an algebraic integer if the minimal monic polynomial of over is in .
- is an algebraic integer if is a finitely generated -module.
- is an algebraic integer if there exists a finitely generated -submodule such that .
Algebraic integers are a special case of integral elements of a ring extension. In particular, an algebraic integer is an integral element of a finite extension .
Read more about this topic: Algebraic Integer
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