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Divisor (of an integer or of a polynomial)

From Encyclopedia of Mathematics - Reading time: 1 min

2020 Mathematics Subject Classification: Primary: 13A05 [MSN][ZBL]

For other meanings of the term 'Divisor' see the page Divisor (disambiguation)

A divisor of an integer a is an integer b which divides a without remainder. In other words, a divisor of the integer a is an integer b such that, for a certain integer c, the equality a=bc holds. A proper divisor or an aliquot divisor of a is a natural number divisor of a other than a itself.

A divisor of a polynomial A(x) is a polynomial B(x) that divides A(x) without remainder (cf. Division).

More generally, in an arbitrary ring R, a divisor of an element aR is an element bR such that a=bc for a certain cR.

If bR is a divisor of aR, one writes b|a.

If a divides b and b divides a, then a and b are associates. If an element a has the property that whenever a=bc, one of b,c is an associate of a, then a is irreducible. For polynomials, see Irreducible polynomial; for integers, the traditional terminology is prime number.

References[edit]

  • David Sharpe, Rings and Factorization Cambridge University Press (1987) ISBN 0-521-33718-6 Zbl 0674.13008

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