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# algebraic integer math. - algebraischer Integer

Examples/ definitions with source references
ALGEBRAIC INTEGER - [math] algebraischer Integer n

Def. algebraic integer
Algebraic number - A complex (sometimes, real) number that is a root of a polynomial … with rational coefficients, not all of which are zero. If α is an algebraic number, then, among all polynomials with rational coefficients and α as a root, there exists a unique polynomial ϕ(x) of lowest degree with leading coefficient equal to one, which is therefore irreducible …. It is called the irreducible, or minimal, polynomial of the algebraic number α. …
An algebraic number is called an algebraic integer if all the coefficients of its minimal polynomial are rational integers. For instance, i
and 1 + √2 are algebraic integers, being roots of the polynomials x^2 + 1 and x^2 − 2x − 1.
Quelle: Encyclopedia of Mathematics (Wiki von Springer / European Mathematical Society)

Def. algebraic integer
An algebraic number is an algebraic integer if it is a root of some monic polynomial f(x) ∈ ℤ[x] (i.e., a polynomial f(x) with integer coefficients and leading coefficient one).

Def. algebraischer Integer
Ein algebraischer Integer ist eine Nullstelle α ∈ ℂ eines Polynoms x^n + b_1 x^(n-1) + … + b_n mit b_1, …, b_n ∈ ℤ
Authorrthappe (1142163) 22 Jun 17, 14:53
Comment Bin mir nicht ganz sicher, aber gebräuchlicher scheint zu sein "ganzalgebraische Zahl" oder "ganze algebraische Zahl":https://de.wikipedia.org/wiki/Algebraische_Zahlhttp://math.johoelken.de/algebra/Vortrag_ganz...  (Definition 7.1)http://de.academic.ru/dic.nsf/dewiki/494317u.a. ...
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## ganz(e) algebraische Zahl

Context/ examples
Def. ganze algebraische Zahl, ganz algebraische Zahl, eine algebraische Zahl, deren normiertes (d..h mit höchstem Koeffizienten 1) Minimalpolynom ganzzahlige (genauer: ganzrationale) Koeffizienten hat.

Comment
Der "algebraische Integer" kommt in Google Books tatsächlich auffallend selten vor, seltener als die "ganze algebraische Zahl" (mit Varianten), wenn er es auch in das eingangs zitierte Wörterbuch (Dorians) geschafft hat.
#2Authorrthappe (1142163) 22 Jun 17, 17:06

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