Difference between revisions of "Divisor (of an integer or of a polynomial)"
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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==== | ||
| + | * 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
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=25206