## Explicit bounds for primality testing and related problems.(English)Zbl 0701.11075

In his very readable book [“Analytic methods in the analysis and design of number-theoretic algorithms” (MIT Press, Cambridge 1985; Zbl 0572.10001)] the author found a constant for Ankeny’s Theorem that if G is a proper subgroup of $$({\mathbb{Z}}/m{\mathbb{Z}})^*$$, then the smallest integer w not in G must exceed 2 log$${}^ 2m$$. Here he extends his methods to algebraic number fields according to J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko [Invent. Math. 54, 271-296 (1979; Zbl 0401.12014)].
Here, analogously, the least prime ideal which does not split in E/K (of absolute discriminant $$\Delta$$, and conductor $${\mathfrak f})$$ has norm less than 3 log$${}^ 2(\Delta^ 2N{\mathfrak f})$$. If we insist that w be relatively prime to m, then the multiplier “2” should be replaced by “3”. Also if the ideal is to be relatively prime to $${\mathfrak f}$$ then the “3” is replaced by a “12”, while if the ideal is to also completely split, the constant becomes “18”. All constants may be replaced by $$1+o(1),$$ and tables support this latter result numerically.
Reviewer: H.Cohn

### MSC:

 11Y11 Primality 11R29 Class numbers, class groups, discriminants 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11A15 Power residues, reciprocity 11R44 Distribution of prime ideals

### Citations:

Zbl 0572.10001; Zbl 0401.12014
Full Text:

### References:

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