Died: 3 May 1988 in Milan, Italy

**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:-

Seidenberg contributed important research to commutative algebra, algebraic geometry, differential algebra, and the history of mathematics. In 1945 he publishedHis career included a Guggenheim Fellowship[awarded1953], 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's papers on differential algebra include... 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".

[Throughout his career Seidenberg published important papers on the history of mathematics. For exampleSeidenberg]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 that0be the only differential polynomial over F vanishing identically on F.

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:-

In the paper he gives conditions on the ringWhen 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?

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*

**November 2004**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Seidenberg.html]