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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:

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