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algebraic integer[¦al·jə¦brā·ik ′in·tə·jər]
a number that is a root of an equation of the form xn + a1xn – 1 + · · · + an = 0, where a1,..., an 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.