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TWO PROBLEMS CONCERNING IRREDUCIBLE ELEMENTS IN RINGS OF INTEGERS OF NUMBER FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 1 / August 2017
- Published online by Cambridge University Press:
- 02 March 2017, pp. 44-58
- Print publication:
- August 2017
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- Article
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Let
$K$ be a number field with ring of integers 
$\mathbb{Z}_{K}$. We prove two asymptotic formulas connected with the distribution of irreducible elements in 
$\mathbb{Z}_{K}$. First, we estimate the maximum number of nonassociated irreducibles dividing a nonzero element of 
$\mathbb{Z}_{K}$ of norm not exceeding 
$x$ (in absolute value), as 
$x\rightarrow \infty$. Second, we count the number of irreducible elements of 
$\mathbb{Z}_{K}$ of norm not exceeding 
$x$ lying in a given arithmetic progression (again, as 
$x\rightarrow \infty$). When 
$K=\mathbb{Q}$, both results are classical; a new feature in the general case is the influence of combinatorial properties of the class group of 
$K$.
