Celebratio Mathematica

S. Mac Lane and O. F. G. Schilling: “Infinite number fields with Noether ideal theories,” Am. J. Math. 61 : 3 (July 1939), pp. 771–782. MR 0000019 JFM 65.0093.01 Zbl 0021.20104 article

The abstract development of many different types of ideal theories in general rings, at the hands of W. Krull [1935] and other students of Emmy Noether suggests a natural converse problem: Can one characterize constructively the set of all rings in which a given type of ideal theory holds? In particular, in what rings does the complete Noether (or classical) prime ideal decomposition theory hold? For this second problem we obtain here a partial answer: if the elements of such a ring are integral in every valuation of the ring, then the ring is necessarily the set of algebraic integers from a classical (finite) algebraic number field or from a certain type of infinite algebraic number field. The other classical case of algebraic function fields is also considered, as well as the closely related problem of finding those fields for which every valuation is discrete.

S. MacLane and O. F. G. Schilling: “Zero-dimensional branches of rank one on algebraic varieties,” Ann. Math. (2) 40 : 3 (July 1939), pp. 507–520. MR 0000158 JFM 65.1140.02 Zbl 0023.29402 article

S. MacLane and O. F. G. Schilling: “Normal algebraic number fields,” Proc. Natl. Acad. Sci. U.S.A. 26 : 2 (February 1940), pp. 122–126. Brief outline of results presented in an article in Trans. Am. Math. Soc. 50:2 (1941). MR 0001232 JFM 66.0122.01 Zbl 0024.01201 article

In this note we present a brief outline of investigations on the generalization of class field theory to arbitrary normal fields. A complete treatment of the results will be submitted for publication to the Transactions of the American Mathematical Society. In the sequel \( k \) denotes an algebraic number field of finite degree over the field of all rational numbers. Class field theory [Chevalley 1936], [Weil 1936; 1938], deals with the description of abelian extensions over \( k \) in terms of classes of ideals in \( k \). In other words, it is the analog of the theory of multiplicative functions on algebraic Riemann surfaces. It is possible to describe the class groups of the classical theory by means of the totality of all \( p \)-adic extensions \( k_p \) of \( k \); \( p \) denoting the prime divisors of \( k \).

S. Mac Lane and O. F. G. Schilling: “Normal algebraic number fields,” Trans. Am. Math. Soc. 50 : 2 (September 1941), pp. 295–384. A brief outline of these results was published in Proc. Natl. Acad. Sci. U.S.A. 26:2 (1940). MR 0005108 JFM 67.0981.03 Zbl 0061.06102 article

S. MacLane and O. F. G. Schilling: “A formula for the direct product of crossed product algebras,” Bull. Am. Math. Soc. 48 : 2 (1942), pp. 108–114. MR 0006152 Zbl 0061.06103 article

S. Mac Lane and O. F. G. Schilling: “A general Kummer theory for function fields,” Duke Math. J. 9 : 1 (1942), pp. 125–167. MR 0006169 Zbl 0061.06101 article

S. MacLane and O. F. G. Schilling: “Groups of algebras over an algebraic number field,” Am. J. Math. 65 : 2 (April 1943), pp. 299–308. MR 0007752 Zbl 0060.07905 article

In this note we shall discuss some new aspects of the theory of normal simple algebras over an algebraic number field. The algebras considered are split by a finite normal extension which is the join of two normal subfields. Moreover, the ramifications of the algebras are to be prime to the discriminant of the given normal field. We propose to find conditions insuring that the group of algebras split by the top field be the join of the groups of algebras split by the given component fields. This query leads to certain curious facts concerning the associated Galois groups. One of our results states that the group of algebras for any abelian field is always the join of the group of algebras split by any set of subfields whose join is the top field. In the general case we have succeeded in finding a partial description of the Galois groups in terms of the condition on the associated algebras. The method of proof of our theorems consists in a reduction of problems on algebras to problems on groups by means of the theory of invariants of algebras.

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Saunders Mac Lane

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