### Abstract

Let k be a differential field of characteristic 0, and Ω be a universal extension of k. Suppose that the field of constants k_{0} of k is algebraically closed. Consider the following differential polynomial of the first order over k in a single indeterminate y: T(y) = (y′)^{2} - λS (y; k) λ ∈ k; λ ≠ 0; here S(y; k) = y(1 - y)(1 - κ^{2}y); κ ∈ k; k^{2} ≠ 0, 1; k′ = 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: (i) z is transcendental over k*; (ii) the field of constants of k* is the same as k_{0}; (iii) 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.

Original language | English |
---|---|

Pages (from-to) | 191-197 |

Number of pages | 7 |

Journal | Pacific Journal of Mathematics |

Volume | 74 |

Issue number | 1 |

Publication status | Published - 1978 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*74*(1), 191-197.

**Transcendental constants over the coefficient fields in differential elliptic function fields.** / Nishioka, Keiji.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 74, no. 1, pp. 191-197.

}

TY - JOUR

T1 - Transcendental constants over the coefficient fields in differential elliptic function fields

AU - Nishioka, Keiji

PY - 1978

Y1 - 1978

N2 - 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 differential polynomial of the first order over k in a single indeterminate y: T(y) = (y′)2 - λS (y; k) λ ∈ k; λ ≠ 0; here S(y; k) = y(1 - y)(1 - κ2y); κ ∈ k; k2 ≠ 0, 1; k′ = 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: (i) z is transcendental over k*; (ii) the field of constants of k* is the same as k0; (iii) 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.

AB - 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 differential polynomial of the first order over k in a single indeterminate y: T(y) = (y′)2 - λS (y; k) λ ∈ k; λ ≠ 0; here S(y; k) = y(1 - y)(1 - κ2y); κ ∈ k; k2 ≠ 0, 1; k′ = 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: (i) z is transcendental over k*; (ii) the field of constants of k* is the same as k0; (iii) 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.

UR - http://www.scopus.com/inward/record.url?scp=84972494000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972494000&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84972494000

VL - 74

SP - 191

EP - 197

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -