algebraic integer

[¦al·jə¦brā·ik ′in·tə·jər]

(mathematics)

The root of a polynomial whose coefficients are integers and whose leading coefficient is equal to 1.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Algebraic Integer

a number that is a root of an equation of the form xⁿ + a₁x^{n – 1} + · · · + a_n = 0, where a₁,..., a_n are integers. For example, is an algebraic integer, since .

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 , where m and n are ordinary integers, are discussed in IDEAL.

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References in periodicals archive

It was proved in [1] that if N C Fin([beta]) then [beta] is a Pisot number, that is, a real algebraic integer greater than 1 with all conjugates strictly inside the unit circle.

Arithmetics on beta-expansions with Pisot bases over [F.sub.q] (([x.sup.-1]))

Algebraic integer (AI) encoding [11] has been proposed for addressing this problem.

Asynchronous realization of algebraic integer-based 2D DCT using achronix speedster SPD60 FPGA

Let [theta] be an algebraic integer, and f (x) the minimal polynomial of 9 over Q.

On the ring of integers of real cyclotomic fields

The field F will be called norm-Euclidean if, for every pair ([alpha],[beta]) of algebraic integers in F with [beta] [not equal to] 0, there exists an algebraic integer [gamma] in F such that [absolute value of ([N.sub.F]([alpha] - [gamma][beta]))] < [absolute value of ([N.sub.F]([beta]))].

Non-norm-euclidean fields in basic [Z.sub.l]-extensions

Then ([GAMMA], S) has the exponential growth rate [tau] = lim [sup.sub.k[right arrow][infinity]] [kth root of [a.sub.k]] which is a real algebraic integer bigger than 1 ([5]).

On the growth rate of ideal coxeter groups in hyperbolic 3-space

Let [theta] be an algebraic integer in [k.sub.1] satisfying the following conditions:

On Redei's dihedral extension and triple reciprocity law

Therefore [omega] > 1, the reciprocal of R, becomes a real algebraic integer whose conjugates have moduli less than or equal to the modulus of !.

On the growth of hyperbolic 3-dimensional generalized simplex reflection groups

Let n be a divisor of [q.sup.*], let [L.sub.n] be a subfield of K = Q([zeta]) with [[L.sub.n]: Q]= n and let [O.sub.n] be the algebraic integer ring of [L.sub.n].

The example by Stephens: dedicated to the memory of Professor Goro Azumaya

Dubickas, Additive Hilbert's Theorem 90 in the ring of algebraic integers, Indag.

Rational quotients of two linear forms in roots of a polynomial

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.

Capacity Theory With Local Rationality: The Strong Fekete-Szego Theorem on Curves

On the other side, algebraic integers lying in an extension field of the rationals are generated with the same linear polynomials.

Massively distributed computing and factoring large integers