Number Theory [Paperback]

From the reviews: ...a fine book [...] When it appeared in 1949 it was a pioneer. Now there are plenty of competing accounts. But Hasse has something extra to offer.[...] Hasse proved that miracles do happen in his five beautiful papers on quadratic forms of 1923-1924. [...]It is trite but true: Every number-theorist should have this book on his or her shelf. --Irving Kaplansky in Bulletin of the American Mathematical Society, 1981

From the reviews:
...a fine book ... treats algebraic number theory from the valuation-theoretic viewpoint. When it appeared in 1949 it was a pioneer. Now there are plenty of competing accounts. But Hasse has something extra to offer. This is not surprising, for it was he who inaugurated the local-global principle (universally called the Hasse principle). This doctrine asserts that one should first study a problem in algebraic number theory locally, that is, at the completion of a vaulation. Then ask for a miracle: that global validity is equivalent to local validity. Hasse proved that miracles do happen in his five beautiful papers on quadratic forms of 1923-1924. ... The exposition is discursive. ... It is trite but true: Every number-theorist should have this book on his or her shelf.
(Irving Kaplansky in Bulletin of the American Mathematical Society, 1981)Part I. The Foundations of Arthmetic in the Rational Number Field:Chapter 1 Prime DecompositionChapter 2 DivisibilityChapter 3 CongruencesChapter 4 The Structure of the Residue Class Ring mod m and the Reduces Residue Class Group mod m.Chapter 5 Quadratic ResiduesPart II. The Theory of Valued FieldsChapter 6 The Fundamental Concepts Regarding ValuationsChapter 7 Arithmetic in a Discrete Valued FieldChapter 8 The Completition of a Valued FieldChapter 9 The Completition of a Discrete Valued Field. The p-adic Number FieldsChapter 10 The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class FieldChapter 11 Prolongation ol³.