Algebraic integer

Integer algebraic numbers are the complex (and, in particular, real ) roots of polynomials with integer coefficients and with a leading coefficient equal to unity.

With respect to the addition and multiplication of complex numbers, integer algebraic numbers form a ring $\Omega$ ${\ displaystyle \ Omega}$ $\ Omega$ . Obviously $\Omega$ ${\ displaystyle \ Omega}$ $\ Omega$ is a subring of the algebraic number field and contains all ordinary integers.

Let be $u$ ${\ displaystyle u}$ $u$ Is some complex number. Consider the ring $\mathbb {Z} [u]$ ${\ displaystyle \ mathbb {Z} [u]}$ ${\ mathbb {Z}} [u]$ generated by adding $u$ ${\ displaystyle u}$ $u$ to the ring of ordinary integers $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ $\ mathbb {Z}$ . It is formed by all sorts of meanings. $f(u)$ ${\ displaystyle f (u)}$ $f (u)$ where $f(z)$ ${\ displaystyle f (z)}$ $f (z)$ Is a polynomial with integer coefficients. Then the following criterion holds: the number $u$ ${\ displaystyle u}$ $u$ is an integer algebraic number if and only if $\mathbb {Z} [u]$ ${\ displaystyle \ mathbb {Z} [u]}$ ${\ mathbb {Z}} [u]$ Is a finitely generated abelian group .

Content

Examples of Algebraic Integers

Gaussian integers .
The roots of unity - the roots of the polynomial $x^{n}-1$ ${\ displaystyle x ^ {n} -1}$ over the field of complex numbers.

Properties

All rational numbers in $\Omega$ ${\ displaystyle \ Omega}$ are integers . In other words, not a single irreducible fraction $m/n$ ${\ displaystyle m / n}$ with a denominator greater than unity, there cannot be an integer algebraic number.
For every algebraic number $u$ ${\ displaystyle u}$ there is a natural number $n$ ${\ displaystyle n}$ such that $n u$ ${\ displaystyle nu}$ Is an integer algebraic number.
A root of any degree from an integer algebraic number is also an integer algebraic number.

History

The theory of integer algebraic numbers was created in the 19th century by Gauss , Jacobi , Dedekind , Kummer and others. The interest in it was, in particular, caused by the fact that historically this structure was the first in mathematics, where an ambiguous decomposition into simple factors was discovered. Classic examples built by Kummer; say, in a subring of algebraic integers of the form $a+b{\sqrt {-5}}$ ${\ displaystyle a + b {\ sqrt {-5}}}$ 2 decompositions take place:

6=2\cdot 3=(1+{\sqrt {-5}})\cdot (1-{\sqrt {-5}})

{\ displaystyle 6 = 2 \ cdot 3 = (1 + {\ sqrt {-5}}) \ cdot (1 - {\ sqrt {-5}})}

,

in both cases, all factors are simple , that is, indecomposable in this subring.

The study of this problem led to the discovery of important concepts of ideal and simple ideal , in the structure of which the decomposition into simple factors became possible to determine unambiguously.

Literature

C. Airland, M. Rosen. A classic introduction to modern number theory. Translation from English by S.P. Demushkin edited by A.N. Parshin. M .: Mir, 1987, chapter 6.
Borevich Z. I., Shafarevich I. P. Number theory. M .: Nauka, 3rd ed., 1985 .-- 504 p.
Van der Waerden B. L. Algebra. M.: Mir, 1975, chapter 17: Entire algebraic elements.
Hecke E. Lectures on the theory of algebraic numbers, trans. with him., M. - L., 1940.
Gelfond A.O. Transcendental and algebraic numbers, M., 1952.
Postnikov M. M. Introduction to the theory of algebraic numbers