## Chebyshev’s method for number fields.(English)Zbl 1007.11069

Let $$\alpha$$ be an irrational algebraic integer such that $${\mathbb{Q}}(\alpha) / {\mathbb{Q}}$$ is Galois. If $$S$$ is the set of primes splitting in $${\mathbb{Q}}(\alpha)$$ then $\sum_{p \in S,\;p \leq x} 1\gg \frac{x^{1/d}}{\log x}, \quad \text{where} \quad d= [{\mathbb{Q}}(\alpha) : {\mathbb{Q}}].$ The proof uses binomial coefficients and extends Chebyshev’s classical approach.

### MSC:

 11R44 Distribution of prime ideals 11N05 Distribution of primes
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### References:

 [1] Chebyshev, P.L., Memoire sur les nombres premiers. J. Math. pures et appl.17 (1852), 366-390. [2] Chudnovsky, D., Chudnovsky, G., Applications of Padé approximations to the Grothendieck conjecture on linear differential equations. In Number theory (New York, 1983-84), 52-100, 1135, Springer, Berlin-New York, 1985. · Zbl 0565.14010 [3] Duke, W., Friedlander, J.B., Iwaniec, H., Equidistribution of roots of a quadratic congruence to prime moduli, Ann. of Math.141 (1995), 423-441. · Zbl 0840.11003 [4] Friedlander, J.B., Estimates for Prime Ideals. J. Number Theory, 12 (1980), 101-105. · Zbl 0428.12010 [5] Landau, E., Über die zu einem algebraischen Zahlkörper gehörige Zetafunktion und die Ausdehnung der Tschebyscheffschen Primzahlentheorie auf das Problem der Verteilung der Primideale. Crelle125 (1903), 64-188. · JFM 33.0215.01 [6] Poincaré, H., Extension aux nombres premiers complexes des Théorèmes de M. Tchebicheff. J. Math. Pures Appl. (4) 8 (1892), 25-68. · JFM 24.0171.02 [7] Vaaler, J., Voloch, J.F., The least nonsplit prime in Galois extensions of Q. J. Number Theory, to appear. · Zbl 0963.11066
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