James Ax

James Burton Ax (10 January 1937 – 11 June 2006) was a mathematician who proved several results in algebra and number theory by using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in Number Theory, which was awarded for a series of three joint papers on Diophantine problems.

James Ax graduated from Peter Stuyvesant High School in New York City and then the Brooklyn Polytechnic University. He earned his Ph.D. from the University of California, Berkeley in 1961 under the direction of Gerhard Hochschild, with a dissertation on The Intersection of Norm Groups. After one year at Stanford University, he joined the mathematics faculty at Cornell University. He spent the academic year 1965–1966 at Harvard University on a Guggenheim Fellowship. In 1969, he moved from Cornell to the mathematics department at Stony Brook University and remained on the faculty until 1977, when he retired from his academic career. In 1970 he was an Invited Speaker at the ICM in Nice with talk Transcendence and differential algebraic geometry. In the 1970s, he worked on the fundamentals of physics, including an axiomatization of space-time and the group theoretical properties of the axioms of quantum mechanics. In the 1980s, he and James Simons founded a quantitative finance firm, Axcom Trading Advisors, which was later acquired by Renaissance Technologies and renamed the Medallion Fund. The latter fund was named after the Cole Prize won by James Ax and the Veblen Prize won by James Simons.

In the early 1990s Ax retired from his financial career and went to San Diego in California, where he studied further on the foundations of quantum mechanics and also attended, at the University of California, San Diego, courses on playwriting and screenwriting. (In 2005 he completed a thriller screenplay entitled Bots.)

The Ax Library in the Department of Mathematics at the University of California, San Diego houses his mathematical books.

• "The intersection of norm groups". Trans. Amer. Math. Soc. 105: 462–474. 1962. doi:10.1090/s0002-9947-1962-0141651-2. MR 0141651. 
• "A field of cohomological dimension 1 which is not C₁". Bull. Amer. Math. Soc. 71: 717. 1965. doi:10.1090/s0002-9904-1965-11354-4. MR 0199177. 
• "On the undecidability of power series fields". Proc. Amer. Math. Soc. 16: 846. 1965. doi:10.1090/s0002-9939-1965-0177890-2. MR 0177890. 
• "Proof of some conjectures on cohomological dimension". Proc. Amer. Math. Soc. 16: 1214–1221. 1965. doi:10.1090/s0002-9939-1965-0188263-0. MR 0188263. 
• "On the units of an algebraic number field". Illinois Journal of Mathematics. 9 (4): 584–589. 1965. MR 0181630. 
• "Solving diophantine problems modulo every prime". Annals of Mathematics, Ser. 2. 85 (2): 161–183. 1967. doi:10.2307/1970438. 
• "The elementary theory of finite fields". Annals of Mathematics, Ser. 2. 88 (2): 239–271. 1968. doi:10.2307/1970573. 
• "Zeros of polynomials over local fields—The Galois action". Journal of Algebra. 15 (3): 417–428. 1970. doi:10.1016/0021-8693(70)90069-4. 
• "On Schanuel's conjectures". Annals of mathematics, Ser. 2. 93 (2): 252–268. 1971. doi:10.2307/1970774.