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The theory of irrationalities of the third degree. (Russian) Zbl 0061.09001

Tr. Mat. Inst. Steklova 11, 340 p. (1940).
This outstanding but eccentric monograph should be in the hands of all interested in algebraic fields of degree \(3, 4\). It is to be hoped that it will be reprinted and made more widely available. It presupposes little knowledge of general theory. The ground field is always the rational field.
Chapter I gives a novel interpretation of algebraic numbers. A number \(\alpha\) in a field \(K\) of degree \(n\) is represented by the point \((\alpha_1, \ldots, \alpha_n)\) in \(n\)-dimensional space; so \(\alpha+\beta\), \(\alpha\beta\) are represented by \((\alpha_1+ \beta_1,\ldots, \alpha_n+ \beta_n)\) and \((\alpha_1 \beta_1,\ldots, \alpha_n \beta_n)\), respectively. The integers of \(K\) form a lattice with a further “multiplicative” property. Properties of algebraic fields are reflected in properties in the corresponding “multiplicative lattices” and a geometric interpretation of Galois theory is given.
Chapter II deals with cubic fields. It treats (i) Voronoi’s determination of the base for the integers, (ii) the factorization of ideals, (iii) the problem whether a given cubic irrational is a perfect power (very neat this), (iv) an effective algorithm for finding whether two cubic equations define the same cubic field (“the inverse Tschirnhausen problem”) and, as an application whether two cubic binary forms are equivalent, (v) a relation between rings in a cubic field.
The first part of Chapter III deals with the tabulation and classification of cubic and quartic fields. This is, in the reviewer’s opinion, one of the less successful parts – the detailed geometrical treatment obscures the essential simplicity of the appeal to Minkowski’s convex body theorem. Instead of dealing with the \(n\)-dimensional lattice of all integers in the relevant field the authors consider the \((n-1)\)-dimensional lattice of all \(\alpha-n^{-1}S(\alpha)\) where \(\alpha\) is an integer, \(S(\alpha)\) its spur (trace) and \(n\) the degree. There is no discussion of the relative merit of the two alternative procedures: it would be interesting to have the authors’ reasons for avoiding the most obvious one [the \(n\)-dimensional approach was used for by J. Mayer, S. B. Akad. Wiss. Wien, Math.-Naturw. Kl., Abt. IIa 138, 733–742 (1929; JFM 55.0104.05), whose work was apparently unknown to the authors]. Also there is no mention of class-field theory which permits a rapid construction of the list of discriminants, especially for \(n=3\).
The second half of Chapter III deals with the construction of cubic fields \(K\) from the “supporting” quadratic fields \(k\) (\(k\) is the quadratic subfield of the normal field of degree 6 containing \(K\)) and similarly the construction of quartic fields from the “supporting” cubic fields. The discussion is “geometrical” and there is no mention of class-field theory [cf. H. Hasse, Math. Z. 31, 565–582 (1930; JFM 56.0167.02); correction 31, 799 (1930)].
Chapter IV gives a geometrical interpretation of Voronoi’s algorithm for finding the units in cubic rings.
Chapter V discusses the Thue-Siegel theorem as it applies to cubic fields. It also improves some results of Siegel an the number of representations of a number by a cubic form: if \(f(x, y)\) is an irreducible binary cubic with integer coefficients and \(k\) is fixed the number of integer solutions of \(| f(x,y)|\leq k\) is at most 15 except for a finite number of forms \(f(x, y)\).
Chapter VI is devoted to the representation of integers by binary cubic forms using the units of the relevant cubic fields. It is shown that in some cases, as \(ax^3+by^3 = 1\), all the solutions may be obtained. Further the number of representations of 1 by an integral binary cubic field of negative discriminant is at most 3; except that there are 5 representations by the form of discriminant \(-23\) and 4 representations by the forms of discriminants \(-31\), and \(-44\). The proofs use Delone’s “algorithm of exponentiation” (Algorithmus der Erhöhung) (cf. T. Nagell [Math. Z. 28, 10–29 (1928; JFM 54.0174.02)], and B. N. Delone, Math. Z. 28, 1–9 (1928; JFM 54.0174.01); 31, 27–28 (1929; JFM 55.0098.04)]). The chapter (and the book) ends with Weil’s simple proof of Mordell’s finite basis theorem for curves of genus 1 and a detailed discussion of the special case \(x^3 + y^3 = Az^3\) (cf. D. K. Faddeev [Tr. Mat. Inst. Steklova 5, 25–40 (1934; Zbl 0009.19601)], and now E. S. Selmer [Acta Math. 85, 203–362 (1951; Zbl 0042.26905)].
There is a very useful set of tables, some of them reproduced from other works and some original.
1. Table of all fields generated by cubic equations \(x^3+bx+c\) with \(| b| < 10\), \(| c| < 10\) together with units and class-numbers [L. W. Reid, Diss. Göttingen (1899; JFM 31.0215.02)].
2. Table of class numbers of some purely cubic fields [R. Dedekind, J. Reine Angew. Math. 121, 40–123 (1900; JFM 30.0198.02)], superseded now by [J. W. S. Cassels, Acta Math. 82, 243–273 (1950; Zbl 0037.02701), p. 270].
3. Table of all totally real cubic rings with discriminant \(D\leq 1296\).
4. Table of all non-totally-real cubic rings with discriminant \(-D < 1000\). The field of discriminant 620 generated by \(\alpha^3 + 8\alpha+ 4=0\) (basis \(\alpha^2\), \(\alpha\), 1 unit \(2\alpha+ 1\) class number 1) is omitted. The table is due to G. B. Mathews and W. E. H. Berwick [Proc. Lond. Math. Soc. (2) 10, 46–53 (1912; JFM 42.0243.03)], and was checked by Delone.
5. Table of all totally real quartic fields with discriminant \(\leq 8112\).
6. Table of all quartic fields with two real and two imaginary conjugates and discriminant \(-D\leq 848\).
7. Table of all totally complex quartic fields having a quadratic subfield and with discriminant \(D\leq 1280\). For tables 5, 6, 7 the Galois groups and the quadratic subfields (if any) are given and for 5, 6 a basis of the integers.
8. Table of basic units for all non-totally real cubic rings with discriminant \(-D\leq 379\).
9. Table of units for all cubic fields of \(3a\) where \(a\leq 70\) due to Markoff (not quite superseded by table mentioned in 2.)
10. Table of all representations \(f(x, y)=1\) where \(f(x, y)\) is a binary cubic of discriminant \(-300\leq D < 0\) [B. N. Delone, Math. Z. 28, 1–9 (1928; JFM 54.0174.01); ibid. 31, 1–26 (1929; JFM 55.0722.02)].
11. Table of all binary cubic forms \(x^3+sx^2+qx+n\) with \(s = 0, \pm 1\) and discriminant \(D\), \(-172\leq D < 0\).
12. Table of basic solutions (in the sense of Mordell) for \(x^3+y^3 = A z^3\) and \(0\leq A\leq 50\) (see remarks above concerning end of Chapter VI).

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11R16 Cubic and quartic extensions
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11Rxx Algebraic number theory: global fields
11E76 Forms of degree higher than two
Full Text: MNR