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Difference between revisions of "Divisor (of an integer or of a polynomial)"

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''of an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336901.png" />''
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An integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336902.png" /> which divides the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336903.png" /> without remainder. In other words, a divisor of the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336904.png" /> is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336905.png" /> such that, for a certain integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336906.png" />, the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336907.png" /> holds. A divisor of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336908.png" /> is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d0336909.png" /> that divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d03369010.png" /> without remainder (cf. [[Division|Division]]). More generally, in an arbitrary ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d03369011.png" /> a divisor of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d03369012.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d03369013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d03369014.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033690/d03369015.png" />.
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''For other meanings of the term 'Divisor' see the page'' [[Divisor (disambiguation)]]
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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.
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A divisor of a polynomial $A(x)$ is a polynomial $B(x)$ that divides $A(x)$ without remainder (cf. [[Division|Division]]).  
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More generally, in an arbitrary [[Ring|ring]] $R$, a divisor of an element $a \in R$ is an element $b\in R$ such that $a=bc$ for a certain $c\in R$.
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If $b\in R$ is a divisor of $a\in R$, one writes $b | a$.
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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]]. 
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====References====
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* David Sharpe, ''Rings and Factorization'' Cambridge University Press (1987) {{ISBN|0-521-33718-6}} {{ZBL|0674.13008}}

Latest revision as of 08:06, 26 November 2023

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 $a \in R$ is an element $b\in R$ such that $a=bc$ for a certain $c\in R$.

If $b\in R$ is a divisor of $a\in R$, 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

  • David Sharpe, Rings and Factorization Cambridge University Press (1987) ISBN 0-521-33718-6 Zbl 0674.13008
How to Cite This Entry:
Divisor (of an integer or of a polynomial). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divisor_(of_an_integer_or_of_a_polynomial)&oldid=16756
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article