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Field arithmetic. Revised by Moshe Jarden. 3rd revised ed. (English) Zbl 1145.12001
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11. Berlin: Springer (ISBN 978-3-540-77269-9/hbk). xxiv, 792 p. (2008).
The third edition of this widely appreciated book appears only three years later than the second edition. We refer to Zbl 0625.12001 and Zbl 1055.12003 for comments to the first two editions.
This new edition improves on the previous edition in two ways: first, it removes a number of mathematical inaccuracies and typos, including three or four pages of missing references; second, it reports on five open problems of the second edition which have been solved since.
We list the problems solved (or partially solved), with reference to the solution: Problems 2 and 3: J. Kollár [Isr. J. Math. 161, 89–101 (2007; Zbl 1143.12003), Theorems 1 and 2]; Problem 3: J. Kollár [Isr. J. Math. 162, 235–251 (2007; Zbl 1140.14009), Theorem 1]; Problem 31: L. Bary-Soroker [Math. Ann. 336, No. 4, 949–961 (2006; Zbl 1111.20029)]; Problem 33: the solution comes from a reorganization of the proof of Corollary 28.5.3 suggested by E. Rosen.
At the end of the book a new selection of open problems is included.

12E30 Field arithmetic
12-02 Research exposition (monographs, survey articles) pertaining to field theory
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
12F12 Inverse Galois theory
12Lxx Connections between field theory and logic
14G05 Rational points
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