Integral element

Algebraic integer

Let be a field with , whence is a ring extension. An element is an algebraic integer iff it is integral over , #m/def/ring i.e. it is the root of some polynomial . We denote the ring of algebraic integers in as , which is clearly an Integrally closed domain.

Properties

  1. An algebraic number is an algebraic integer iff its minimal polynomial satisfies .
  2. Every algebraic number is an algebraic integer divided by some integer.
Proof of 1–2

Clearly if then is integral over . Now suppose is integral over , so it is the root of some monic polynomial . Let denote the roots of . Since , it follows for all , so are each algebraic integers. Since for some algebraic multiplicities has coëfficients which are the products of algebraic integers, its coëfficients are themselves algebraic integers, and thus by ^P1, proving ^P1.

Let

be the minimal polynomial of where . Then for some whence

so is an algebraic integer.

Other results

Special case

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