We use the nonstandard methods of Chapter 14 to give a new criterion for a field K to be Hilbertian: There exists a nonstandard element t of an enlargement K* of K such that t has only finitely many poles in K(t) s ∩K*. From this there results a second and uniform proof (Theorem 15.3.4) that the classical Hilbertian fields are indeed Hilbertian. In addition, a formal power series field, K0((X1,…,X n )) of n≥2 variables over an arbitrary field K0, is also Hilbertian (Example 15.5.2).
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Nonstandard Approach to Hilbert’s Irreducibility Theorem. In: Field Arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77270-5_15
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DOI: https://doi.org/10.1007/978-3-540-77270-5_15
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