Two books by Wilhelm Specht

Here we look at brief extracts from reviews of two books by Wilhelm Specht, namely Gruppentheorie (1956) and Algebraische Gleichungen mit reellen oder komplexen Koeffiziente (1958).

1. Gruppentheorie (1956), by Wilhelm Specht.
1.1. Review by: Bernhard Neumann.
The Mathematical Gazette 42 (340) (1958), 151-154.

[This book] is not a text-book, nor a reference book. The author's stated aim is to "make a selection from the gigantic abundance of results that will let the reader recognize the beauty of the theory and its manifold methods". In this he fails: no reader will recognise the beauty of group theory in this book unless he is already well familiar with it. The style is extremely condensed, and full use is made of symbols and formulas. On many pages the formulas outweigh the text, and the demands made on the reader are invariably very high: he has to extract significance from pages after pages of intricate formal arguments, without any guidance as to the motivation and organisation of the theory. The emphasis is so even throughout that it is difficult to separate the important from the unimportant. ... Specht's book has crystallised from a lone, patient and thorough study of the modern literature on the subject. This is not to say that Specht's book lacks originality; the author has devised a novel and systematic notational scheme, which has brought in its train a generalisation and unification of many diverse results, and has thus thrown new light on some aspects of the theory.

1.2. Review by: R H Bruck.
Mathematical Reviews, MR0080091 (18,189b).

This book was at first so impressive in its wealth of material that the reviewer regarded it as a complete encyclopedia on the subject. Soberer thoughts showed otherwise; it stands on a par with the well-known texts by Burnside, Speiser and Zassenhaus and the more recent volumes of Kurosh, but it does not supersede any of them. Rather than compare and contrast these books in detail, we shall quote some timely words from the Foreword: "An austere selection of material is, for all that, necessarily a function of personal taste; this truth must console the connoisseur if he misses this or that which lies in his heart."

2. Algebraische Gleichungen mit reellen oder komplexen Koeffizienten (1958), by Wilhelm Specht.
2.1. Review by: Morris Marden.
Bull. Amer. Math. Soc. 66 (2) (1960), 63-64.

Though entitled Algebraic equations, Specht's article (as it is explained in its introduction) deals with material usually called the "analytic theory of polynomials" or the "geometry of the zeros." ... Important theorems on symmetric functions of the zeros, quadratic and hermitian forms and Kronecker's method of characteristics are reviewed. Also included are theorems concerning the characteristic roots of matrices ... Specht's article [is] a well-organized survey of the analytic theory of polynomials. It should prove to be a valuable up-to-date reference source to all who work regularly or even occasionally in this field.

2.2. Review by: Jean Dieudonné.
Mathematical Reviews, MR0106274 (21 #5008).

The author reports on the subject called "Analytic theory of polynomials'', i.e. the properties of polynomials considered as special analytic functions of one complex variable, centering especially on the dependence of the zeros of a polynomial on its coefficients ... There is no attempt at complete coverage of the relevant literature, but the general methods are clearly delineated and the booklet is a valuable adjunction to the monographs on the subject.

JOC/EFR April 2016

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