# THE FOUNDATION OF COMPUTER ALGEBRA

## HERMANN, GRETE.

# Die Frage der endlich vielen Schritte in der Theorie der Polynomideale.

(Berlin, Julius Springer, 1926).

8vo. Without wrappers. Extracted from "Mathematische Annalen. Begründet 1868 durch Alfred Clebsch und Carl Neumann. 95. Band". Pp. 735-788.

*First publication of Hermann's seminal paper (her doctoral thesis) which founded computer algebra. It first established the existence of algorithms - including complexity bounds - for many of the basic problems of abstract algebra, such as ideal membership for polynomial rings. Hermann's algorithm for primary decomposition is still in use today. The paper anticipates the birth of computer algebra by 39 years.*

"[The paper] is an intriguing example of ideas before their time. While computational aspects of mathematics were more fashionable before the abstractions of the twentieth century took hold, mathematicians of that time certainly knew nothing of computers nor of today's idea of what an algorithm is. The significance of the paper can be found on the first page, where we find (in translation):The claim that a computation can be found in finitely many steps will mean here that an upper bound for the number of necessary operations for the computation can be specified. Thus it is not enough, for example, to suggest a procedure, for which it can be proved theoretically that it can be executed in finitely many operations, if no upper bound for the number of operations is known. The fact that the author requires an upper bound suggests that there must exist an actual procedure or algorithm for doing computations. We see in this paper the first examples of procedures (with upper bounds given) for a variety of computations in multivariate polynomial ideals. Thus we have here a paper anticipating by 39 years the birth of computer algebra". (ACM SIGSAM Bulletin, Volume 32. 1998).

Not in Hook & Norman.

Order-nr.: 44771