No CrossRef data available.
On multiple curves. III.
Published online by Cambridge University Press: 24 October 2008
Extract
A curve Γ has been defined as a normal multiple of a curve C if it contains an involution In, of order n, which has the properties: (1) the sets of In are in (1–1) correspondence with the points of C; (2) each set In consists of n points which are distinct in the birational sense; (3) In is generated by an Abelian group G of order n of birational transformations of Γ into itself. We shall denote the field of complex numbers by k, the function field of C by k(C), and the function field of Γ by k(Γ). Conditions (1) and (3) imply that k(Γ) is a commutative normal (i.e. Galois) extension of k(C). The object of this note is to show how, given the curve C and the Abelian group G of order n, we can construct a curve Γ with the properties (1), (2), (3).
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 42 , Issue 1 , February 1946 , pp. 11 - 14
- Copyright
- Copyright © Cambridge Philosophical Society 1946
References
* Hodge, W. V. D., ‘On Multiple Curves, I’, Proc. Cambridge Phil. Soc. 41 (1945).Google Scholar
† Cf. Bliss, G. A., Algebraic Functions (American Math. Soc. Colloquium Publication, 1933), p. 112.Google Scholar
‡ Comessatti, A., Rend. Sem. Mat. Padova, 1 (1930), 1.Google Scholar
* Cf. Baker, H. F., Abel's Theorem and the Allied Theory, Cambridge (1897). Chap. XIII.Google Scholar
† Albert, A. A., Modern Higher Algebra (Chicago, 1936), 185.Google Scholar
* A. A. Albert, loc. cit. p. 169.