The embedding property (Proposition 17.7.3) for free profinite groups is essential to the primitive recursive procedure for perfect PAC fields with free absolute Galois groups (Theorem 30.6.2). Since this is only a special case of a general result, we focus attention here on PAC fields whose absolute Galois groups have the embedding property: the Frobenius fields. The “field crossing argument” (e.g., the proofs of Lemma 6.4.8, Part G in Section 6.5, Part C of Proposition 16.8.6, Lemma 20.2.2, and Lemma 23.2.1) applies to give an analog, for Frobenius fields, of the Chebotarev density theorem (Proposition 24.1.4). The remainder of the Chapter concentrates on the embedding property. For example, if a profinite group has the embedding property, so does its universal Frattini cover (Proposition 24.3.5). We further show that every profinite group G has a smallest cover E(G) with the embedding property (Proposition 24.4.5). In particular, for G finite, E(G) is finite and unique (Proposition 24.4.6). The construction of E(G) leads to a decision procedure for projective groups with the embedding property (Corollary 24.5.3).
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Frobenius Fields. 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_24
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DOI: https://doi.org/10.1007/978-3-540-77270-5_24
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