Algebraic Integer

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algebraic integer

[¦al·jə¦brā·ik ′in·tə·jər]
The root of a polynomial whose coefficients are integers and whose leading coefficient is equal to 1.

Algebraic Integer


a number that is a root of an equation of the form xn + a1xn – 1 + · · · + an = 0, where a1,..., an are integers. For example, Algebraic Integer is an algebraic integer, since Algebraic Integer.

The theory of algebraic integers developed in the 1830’s and 1840’s as a result of the work of K. Jacobi, F. Eisenstein, and E. Kummer on reciprocity laws of higher degree, on Fermat’s theorem, and on the generalization of the arithmetic of complex integers.

The sum, difference, and product of algebraic integers are algebraic integers; that is, the set of algebraic integers forms a ring. The theory of the divisibility of algebraic integers, however, differs from the theory of the divisibility of ordinary integers. Algebraic integers of the form Algebraic Integer, where m and n are ordinary integers, are discussed in IDEAL.

References in periodicals archive ?
By making use of Euclidean lengths of algebraic integral numbers in a related algebraic integer ring, we discuss the Frobenius expansions of algebraic numbers, theoretically and experimentally show that the multiplier in a scalar multiplication can be reduced and converted into a Frobenius expansion of length approximate to the field extension degree, and then propose an efficient scalar multiplication algorithm.
Rumely's introduction to this exposition on the Fekete-Szego theorem includes a sketch of the basic proof and contextualizes its development with the Robinson theorem on totally real algebraic integers in an interval and Cantor's extension with splitting conditions.
Their topics include divisibility, polynomial congruences, quadratic reciprocity, the geometry of numbers, and algebraic integers.

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