divisibility by prime number
Proof. Suppose that . Then either or . In the latter case we have , and therefore the corollary of Bézout’s lemma gives the result . Conversely, if or , then for example for some integer ; this implies that , i.e. .
Remark 1. The theorem means, that if a product is divisible by a prime number, then at least one of the factor is divisibe by the prime number. Also conversely.
Remark 2. The condition (1) is expressed in of principal ideals as
Here, is a prime ideal of .
|Title||divisibility by prime number|
|Date of creation||2013-03-22 14:48:18|
|Last modified on||2013-03-22 14:48:18|
|Last modified by||pahio (2872)|
|Synonym||divisibility by prime|