Kaup–Kupershmidt equation: Difference between revisions
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In [[mathematics]], an element ''r'' of a [[unique factorization domain]] ''R'' is called '''square-free''' if it is not [[integral domain|divisible]] by a non-trivial square. That is, every ''s'' such that <math>s^2\mid r</math> is a [[unit (algebra)|unit]] of ''R''. | |||
Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit ''r'' can be represented as a product of [[prime element]]s | |||
:<math>r=p_1p_2\cdots p_n</math> | |||
Then ''r'' is square-free if and only if the primes ''p<sub>i</sub>'' are pairwise [[associated element|non-associated]] (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number). | |||
Common examples of square-free elements include [[square-free integer]]s and [[square-free polynomial]]s. | |||
[[Category:Ring theory]] | |||
Revision as of 14:25, 11 January 2014
In mathematics, an element r of a unique factorization domain R is called square-free if it is not divisible by a non-trivial square. That is, every s such that is a unit of R.
Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements
Then r is square-free if and only if the primes pi are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number).
Common examples of square-free elements include square-free integers and square-free polynomials.