Vol. 74, No. 1, 1978

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Transcendental constants over the coefficient fields in differential elliptic function fields

Keiji Nishioka

Vol. 74 (1978), No. 1, 191–197

Let k be a differential field of characteristic 0, and Ω be a universal extension of k. Suppose that the field of constants k0 of k is algebraically closed. Consider the following dlfferential polynomial of the first order over k in a single indeterminate y:

T (y) = (y′)2 − λS(y;κ); λ ∈ k; λ ⁄= 0;


S (y;κ) = y(1 − y)(1 − κ2y); κ ∈ k; κ2 ⁄= 0,1;  κ′ = 0.

Take a generic point z of the general solution of T. Then, z is transcendental over k, and k(z,z) is called a differential elliptic function field.

We prove the following:

Theorem. Let k(z,z) be a differential elliptic function field over k. Then, there exists a finitely generated differential extension field k of k such that the following three conditions are satisfied:

  1. z is transcendental over k;
  2. the field of constants of k is the same as k0;
  3. there exists an element ζ of Ω such that k(z,z) = k(ζ,ζ) and (ζ)2 = 4S(ζ;κ) with the same modulus as κ.

Mathematical Subject Classification 2000
Primary: 12H05
Secondary: 34A20
Received: 24 November 1976
Revised: 20 July 1977
Published: 1 January 1978
Keiji Nishioka