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Abraham Seidenberg studied at the University of Maryland and was awarded his B.A. in 1937. His doctoral studies in algebra were at Johns Hopkins University where his research was supervised by Oscar Zariski. After submitting his Ph.D. thesis Valuation Ideals in Rings of Polynomials in Two Variables he was awarded his doctorate in 1943. In 1945 Seidenberg was appointed as an instructor in mathematics at the University of California at Berkeley. He was promoted rapidly and in 1958 reached the rank of full professor. He retired in 1987 and was made Professor, Emeritus at that time.
Seidenberg married the writer Ebe Cagli. She was born in Ancona, Italy, on 23 February 1915 into a family of Jewish origins. She left Italy with the other members of her family in 1938 after racial persecution and they emigrated to the United States. After a stay in New York she married Seidenberg. Ebe was the author of novels on the exile of the Jews during Fascism. Her brother Corrado Cagli was famed as a painter. Seidenberg and his wife frequently visited Italy. He held a Visiting Professorship at the University of Milan and he gave several series of lectures there. In fact he was in Milan in the middle of giving a lecture series at the time of his death. Ebe Seidenberg died in a clinic in Rome at the age of 87.
M A Rosenlicht, G P Hochschild, and P Lieber in an obituary, describe other features of their colleague Seidenberg's career at Berkeley:-
His career included a Guggenheim Fellowship [awarded 1953], visiting Professorships at Harvard and at the University of Milan, and numerous invited addresses, including several series of lectures at the University of Milan, the National University of Mexico, and at the Accademia dei Lincei in Rome. At the time of his death, he was in the midst of another series of lectures at the University of Milan.
Seidenberg contributed important research to commutative algebra, algebraic geometry, differential algebra, and the history of mathematics. In 1945 he published Valuation ideals in polynomial rings which included results from his doctoral thesis. In the following year he published Prime ideals and integral dependence written jointly with I S Cohen which greatly simplified the existing proofs of the going-up and going-down theorems of ideal theory. An example of one of his papers on algebraic geometry is The hyperplane sections of normal varieties (1950) which has proved fundamental in later advances. He also wrote a book Elements of the theory of algebraic curves (1968). W E Fulton, in a review, describes it as:-
... a well-written text on the theory of algebraic curves. ... [T]he leisurely style, with plenty of motivational discussion, makes it especially useful for an introduction to the subject. Concepts such as plane curve, intersection multiplicity, branch, genus, and linear series are introduced in a concrete, computational way; the necessary abstract algebra is kept in a secondary position whenever possible. Novel features are a chapter on ground fields of positive characteristic and one on "infinitely near points".
Seidenberg's papers on differential algebra include Some basic theorems in differential algebra (characteristic p, arbitrary) (1952) and Some basic theorems in partial differential algebra (1958). Kolchin writes the following in a review of this paper:-
[Seidenberg] reexamines certain known theorems. In the first part he shows that the usual definition of "(differentially) algebraic" is equivalent to one using induction on the number of derivation operators. Certain desired properties follow more easily from the first definition, and others from the second. By including all these properties and the equivalence in one inductive proof, he effects a certain economy. In the subsequent parts he proves that, in a separable differential field extension, every differential transcendence basis is separating, a result previously proved by him in the case of ordinary differential fields; and he also discusses the connection between the condition that every finitely generated extension of a differential field F be simply generated and the condition that 0 be the only differential polynomial over F vanishing identically on F.
Throughout his career Seidenberg published important papers on the history of mathematics. For example Peg and cord in ancient Greek geometry (1959) in which he argues that the whole of Greek geometry had a ritual origin. In The diffusion of counting practices (1960) Seidenberg argues that counting was diffused from one centre and was not discovered again and again as is commonly believed. History of mathematics papers published after he retired include The zero in the Mayan numerical notation (1986) and On the volume of a sphere (1988). In this latter paper Seidenberg compares the methods for calculating the volume of a sphere: in Greek mathematics, namely that by Archimedes; in Chinese mathematics, namely in the Nine Chapters on the Mathematical Art ; in Babylonian mathematics; and in Egyptian mathematics. He argues, as he does in other papers, that there were two traditions in ancient mathematics, see  where this is discussed fully. One was a geometric-constructive traditions and the other an algebraic-computational tradition. These, he claims, originated from a common source prior to Greek, Babylonian, Chinese, and Vedic mathematics. He also argues that the use methods of the Cavalieri type to determine volume go back to this common source. In Geometry and Algebra in Ancient Civilizations Van der Waerden puts forward similar views for which he gives credit to Seidenberg, saying that Seidenberg made him look at the history of mathematics a new way.
We must not suppose that Seidenberg neglected his algebraic research in the latter part of his career. He continued to publish papers such as On the Lasker-Noether decomposition theorem (1984) which asks:-
When does the Lasker-Noether decomposition theorem, which says that an ideal in a commutative Noetherian ring is the intersection of a finite number of primary ideals, hold in a constructive sense?
In the paper he gives conditions on the ring R so that given generators for an ideal in a R[x1, ... , xn] then there is an algorithm to compute generators of the primary ideals and of their associated prime ideals.
M A Rosenlicht, G P Hochschild, and P Lieber end their obituary with these words:-
Those who knew Seidenberg well, including many students, remember his warmth, compassion and integrity. He had a number of very dear friends.
Article by: J J O'Connor and E F Robertson
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