rational algebraic integers
Theorem. A rational number is an algebraic integer iff it is a rational integer.
Proof. . Any rational integer has the minimal polynomial , whence it is an algebraic integer.
. Let the rational number be an algebraic integer where are coprime integers and . Then there is a polynomial
with such that
Multiplying this equation termwise by implies
which says that (see divisibility in rings). Since and are coprime and positive, it follows that . Therefore, .
Title | rational algebraic integers |
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Canonical name | RationalAlgebraicIntegers |
Date of creation | 2013-03-22 19:07:34 |
Last modified on | 2013-03-22 19:07:34 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 5 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 11R04 |
Related topic | MultiplesOfAnAlgebraicNumber |