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The late Helmut Hasse wrote two treatises on number theory. Their first editions appeared in rapid succession in 1949 and 1950. The first, the "blue book", was entitled Zahlentheorie and was published by Akademie-Verlag
(2nd ed. 1963, 3rd ed. 1969). It was a book on algebraic number theory. The second, the "yellow book", was Vorlesungen iiber Zahlentheorie, a book on elementary number theory published by Springer-Verlag (2nd ed. 1964).
Lovers of number theory will now have to be a little careful: both books are yellow. The volume under review is a translation into English of the third edition of the blue book; in moving from Akademie to Springer it changed
color. Beyond that the major change is a recasting of Chapter 16 (on tamely ramified extensions) to remove an error detected by Leicht and Roquette; the rewriting was done by Leicht.
None of the earlier editions was reviewed in this Bulletin. I think a review is still timely, for it is a fine book. It 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 valuation. 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. But I cannot end this paragraph without calling attention to Hasse's eloquent attribution of the key idea to his teacher Hensel; see vol. 209 (1962) of Crelle and an amplification in his preface to volume I of his Mathematische Abhandlungen.
We can now read in English Hasse's disavowal of a "Satz-Beweis" format in favor of a discursive exposition. And indeed the exposition is discursive. The first 100 pages take the reader on a trip through elementary number
theory that reaches quadratic reciprocity in the Hubert product formula version. Then comes a 200-page book within a book on valuation theory. At last, halfway through the book, algebraic number theory begins
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